#417016

order	512
nilpotency class	3
isoclinism	isoclinic class of #398102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1135 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,22)(18,23,21,24);(9,11)(10,14)(12,21,22,18)(13,19)(15,24,20,23)(16,17);(2,5)(6,7)(9,12)(10,15,16,20)(11,18)(13,22)(14,23,17,24)(19,21);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=e,d^2=h,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,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, 324212);