#124204

order	512
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#329
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#376	C₂³
socle	C₂	#7567
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

295 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,13,16,37,17,33,24,51)(4,9)(5,21,22,49)(6,11)(7,12)(8,15,27,41,29,43,28,42)(10,23,34,50)(14,36,39,58,40,55,47,64)(18,44,45,62)(19,31)(20,32)(25,35)(26,38,52,59,54,61,53,60)(30,46,56,63)(48,57);(1,3)(2,8)(4,14)(5,15)(6,24)(7,17)(9,26)(10,33)(11,27)(12,29)(13,34)(16,25)(18,38)(19,47)(20,40)(21,37)(22,43)(23,41)(28,35)(30,55)(31,52)(32,54)(36,56)(39,48)(42,50)(44,58)(45,61)(46,59)(49,51)(53,57)(60,63)(62,64);(1,4,6,19,7,20,25,48)(2,9,11,31,12,32,35,57)(3,14,24,47,17,40,16,39)(5,18,23,46,22,45,50,63)(8,26,27,52,29,54,28,53)(10,30,21,44,34,56,49,62)(13,36,51,64,33,55,37,58)(15,38,41,59,43,61,42,60)

Polycyclic presentation:

2^9|a^2=e,c^2=fh,d^2=g,e^2=h,f^2=i,b^a=bd,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31400);