#11871

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

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

normal subgroups only

This group has 180 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:

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

Polycyclic presentation:

2^8|a^2=e,b^2=h,c^2=g,d^2=gh,e^2=h,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	#130235, #130236, #130285, …
maximal normal subgroup of	#130235, #130236, #130285, …
maximal quotient of	#122777, #122778, #122779, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4859);