#6463633

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1839 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:

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

Polycyclic presentation:

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

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, 6370829);