#6542916

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	#29600	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 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:

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

Polycyclic presentation:

2^9|a^2=fgi,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, 6450112);