#6545998

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(3,7)(4,6)(5,8)(17,18,22,23,21,25,19,26)(20,24,28,30,29,32,27,31);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,22)(18,23,25,26)(20,28,29,27)(24,31,32,30);(10,13)(14,16)(18,23)(19,22)(24,30)(25,26)(27,28)(31,32);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,21,29)(18,24,25,32)(19,27,22,28)(23,30,26,31);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(20,29)(24,32)(27,28)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6453194);