#348540

order	512
nilpotency class	3
isoclinism	isoclinic class of #348538
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1008 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255736);