#7002561

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
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	#38904	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgi,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313864, #72313909, #72313981
maximal normal subgroup of	#72313864, #72313909, #72313981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6909757);