#93949

order	512
nilpotency class	5
isoclinism	isoclinic class of #7414
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1345
commutator	C₂×M₁₆	C₂×C₈
Frattini, cosocle	#1729	C₂²
socle	C₂	#7122
prime radical	C₂⁴	#103
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ #1729 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 203 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)(3,14,16,45,18,46,25,58)(4,19,21,54)(6,11,27,44)(7,13,28,42)(8,30,31,49,34,50,39,15)(9,35,37,23)(12,26,41,29)(17,53,52,64,47,36,51,62)(20,32,55,60,48,33,61,59)(22,38,57,40)(24,43,56,63);(1,3,6,17,7,18,29,47)(2,8,11,32,13,34,12,33)(4,20,22,15,24,48,23,49)(5,25,26,52,28,16,27,51)(9,36,19,46,38,53,43,14)(10,39,41,60,42,31,44,59)(21,55,35,50,56,61,57,30)(37,62,63,45,40,64,54,58)

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,f^a=fi,h^a=hi,c^b=cf,d^b=defhi,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,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, 1145);