#6503058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6410254);