#405637

order	512
nilpotency class	3
isoclinism	isoclinic class of #393963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312833);