#6817878

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,5)(3,7)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,30,29)(20,24,31,28)(21,27,32,25);(1,2)(3,6)(4,5)(7,8)(17,19)(18,23)(20,21)(22,30)(24,27)(25,28)(26,29)(31,32);(9,11,12,16)(10,13,14,15)(17,20,18,24,22,31,26,28)(19,27,29,21,30,25,23,32);(1,3,4,7)(2,6,5,8)(17,18,22,26)(19,23,30,29)(20,28,31,24)(21,25,32,27);(2,5)(6,8)(17,21,18,25,22,32,26,27)(19,28,29,20,30,24,23,31)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313354, #72313674, #72313767
maximal normal subgroup of	#72313354, #72313674, #72313767

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6725074);