#33588

order	256
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,4)(5,6,11,16)(7,12,18,33)(8,21,24,55)(9,25,29,54)(10,30,27,15)(13,36,38,22)(14,39,41,28)(17,43,44,66)(19,49,47,65)(20,52,48,35)(23,56,31,40)(26,42,57,37)(32,58,59,45)(34,62,61,51)(46,67,53,63)(50,64,68,60);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,32)(14,34)(15,35)(16,33)(21,43)(22,45)(23,46)(24,44)(25,49)(26,50)(27,48)(28,51)(29,47)(30,52)(31,53)(36,58)(37,60)(38,59)(39,62)(40,63)(41,61)(42,64)(54,65)(55,66)(56,67)(57,68);(5,8,11,24)(6,13,16,38)(7,17,18,44)(9,26,29,57)(10,31,27,23)(12,32,33,59)(14,40,41,56)(15,42,30,37)(19,50,47,68)(20,53,48,46)(21,54,55,25)(22,39,36,28)(34,63,61,67)(35,64,52,60)(43,65,66,49)(45,62,58,51);(7,18)(12,33)(17,44)(19,47)(20,48)(32,59)(34,61)(35,52)(43,66)(45,58)(46,53)(49,65)(50,68)(51,62)(60,64)(63,67)

Polycyclic presentation:

2^8|a^2=h,b^2=e,c^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^c=fg,g^c=gh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#500752, #500755, #501023, …
maximal normal subgroup of	#500752, #500755, #501023, …
maximal quotient of	#490437, #490443, #490461, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26576);