#124495

order	512
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1530
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#376	C₂³
socle	C₂	#7590
prime radical	C₂³	#410
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 265 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)(3,13,16,39,17,40,28,55)(4,18,21,51)(5,23,26,57)(6,11)(7,12)(8,15,30,47,32,24,31,48)(9,22,34,52)(10,27,36,53)(14,41,43,38,45,63,44,60)(19,37)(20,35)(25,33)(29,59,61,42,62,64,49,46)(50,58)(54,56);(1,3,7,17)(2,8)(4,14,21,45)(5,24,26,15)(6,28,19,16)(9,29)(10,13)(11,30)(12,32)(18,49)(20,46,25,42)(22,43,52,44)(23,55)(27,48,53,47)(31,37)(33,60)(34,62)(35,38)(36,40)(39,57)(41,50)(51,61)(54,64,58,59)(56,63);(1,4,19,52,7,21,6,22)(2,9,11,18,12,34,37,51)(3,14,16,44,17,45,28,43)(5,25,53,54,26,20,27,58)(8,29,30,49,32,62,31,61)(10,35,23,56,36,33,57,50)(13,38,55,63,40,60,39,41)(15,46,48,59,24,42,47,64)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fgh,d^2=gi,e^2=h,f^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,d^b=dg,e^b=efg,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, 31691);