#124457

order	512
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1530
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#376	C₂³
socle	C₂	#7587
prime radical	#3147	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

351 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,13,16,39,18,41,17,40)(4,19,22,52)(5,24,27,57)(6,11)(7,12)(8,15,30,47,31,25,36,49)(9,23,33,53)(10,28,35,54)(14,42,44,60,45,63,48,38)(20,37)(21,34)(26,32)(29,59,50,46,62,64,61,43)(51,58)(55,56);(1,3)(2,8)(4,14)(5,25)(6,16)(7,18)(9,29)(10,13)(11,36)(12,31)(15,27)(17,20)(19,50)(21,46)(22,45)(23,48)(24,39)(26,43)(28,47)(30,37)(32,60)(33,62)(34,38)(35,41)(40,57)(42,51)(44,53)(49,54)(52,61)(55,64)(56,63)(58,59);(1,4,20,53,7,22,6,23)(2,9,11,19,12,33,37,52)(3,14,17,44,18,45,16,48)(5,26,54,55,27,21,28,58)(8,29,36,50,31,62,30,61)(10,34,24,56,35,32,57,51)(13,38,39,63,41,60,40,42)(15,46,47,59,25,43,49,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31653);