#470397

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1196 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=g,c^2=e,d^2=eg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377593);