#33470

order	256
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#442
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,14,18,44)(4,20)(5,10)(6,27)(7,13)(8,17,33,52)(9,23)(11,25)(12,26)(15,45,48,64)(16,43,19,38)(21,35)(22,39)(24,36)(28,41,53,32)(29,34)(30,37)(31,57,62,60)(40,50)(42,49,61,46)(47,63,51,54)(55,58)(56,59);(1,3)(2,8)(4,15)(5,16)(6,28)(7,18)(9,31)(10,32)(11,38)(12,41)(13,33)(14,37)(17,50)(19,26)(20,54)(21,46)(22,57)(23,49)(24,48)(25,52)(27,44)(29,60)(30,53)(34,45)(35,62)(36,63)(39,64)(40,43)(42,55)(47,56)(51,58)(59,61);(1,4,5,23,7,24,26,21)(2,9,12,20,13,35,10,36)(3,15,16,49,18,48,19,46)(6,29,25,56,30,22,50,58)(8,31,41,54,33,62,32,63)(11,39,27,59,40,34,37,55)(14,42,38,64,44,61,43,45)(17,51,28,60,52,47,53,57);(1,5,7,26)(2,10,13,12)(3,16,18,19)(4,21,24,23)(6,25,30,50)(8,32,33,41)(9,20,35,36)(11,37,40,27)(14,43,44,38)(15,46,48,49)(17,28,52,53)(22,56,29,58)(31,54,62,63)(34,55,39,59)(42,64,61,45)(47,60,51,57)

Polycyclic presentation:

2^8|a^2=f,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gh,e^b=egh,f^b=fh,g^b=gh,d^c=dh,e^c=eh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#496131, #496132, #497427, …
maximal normal subgroup of	#496131, #496132, #497427, …
maximal quotient of	#483593, #483594, #483595, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26458);