#6625189

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#25729	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308792, #72308919
maximal normal subgroup of	#72308792, #72308919

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6532385);