#6513695

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1554 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6420891);