#6545488

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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 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)(3,8)(4,6)(5,7)(17,18,23,32,22,26,21,25)(19,27,29,24,28,31,30,20);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(19,28)(20,24)(27,31)(29,30);(10,13)(14,16)(17,19)(18,24)(20,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,24)(18,19,26,28)(21,31,23,27)(25,30,32,29);(11,15)(14,16)(17,21,22,23)(18,25,26,32)(19,29,28,30)(20,27,24,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=fi,e^2=i,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, 6452684);