#93947

order	512
nilpotency class	5
isoclinism	isoclinic class of #7410
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1344
commutator	C₂²×C₈	C₂×C₈
Frattini, cosocle	#1729	C₂²
socle	C₂	#7122
prime radical	C₂⁴	#103
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ #1729 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

202 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defhi,f^b=fi,g^b=gi,d^c=dhi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1143);