#6468369

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1498 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,12,15)(17,18,23,27)(19,28,26,22)(20,24,31,29)(21,25,32,30);(11,12)(14,15)(17,19,22,18,23,26,28,27)(20,30,21,29,31,25,32,24);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,25)(21,22)(23,31)(26,30)(27,29)(28,32);(9,12,13,11)(10,15,16,14)(17,21)(18,25)(19,29)(20,28)(22,31)(23,32)(24,26)(27,30);(1,3,4,7)(2,6,5,8)(20,31)(21,32)(24,29)(25,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375565);