#6460417

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1849 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:

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

Polycyclic presentation:

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

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, 6367613);