#401700

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1610 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:

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

Polycyclic presentation:

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

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, 308896);