#6450332

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(9,10,12,13)(11,14,15,16)(17,18,22,23)(19,27,29,24)(20,25,32,31)(21,26,28,30);(1,3,5,4)(2,6,7,8)(17,19,22,29)(18,24,23,27)(20,30,32,26)(21,25,28,31);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,22,32)(18,25,23,31)(19,28,29,21)(24,26,27,30);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,20)(22,28)(23,30)(24,31)(25,27)(29,32);(17,18,22,23)(19,24,29,27)(20,31,32,25)(21,26,28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357528);