#11869

order	256
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1519
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂×M₁₆	C₂³
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 55 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^8|a^2=e,b^2=h,c^2=gh,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,d^c=dh,e^d=eh

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#130238, #130239, #130286, …
maximal normal subgroup of	#130238, #130239, #130286, …
maximal quotient of	#122701, #122702, #122703, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4857);