#470998

order	512
nilpotency class	3
isoclinism	isoclinic class of #460096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1027 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)(3,6)(4,5)(7,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,20,22)(21,24);(9,11)(10,14)(12,20,21,18)(13,19)(15,23,24,22)(16,17);(2,5)(6,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,24)(17,23)(19,20);(1,3,4,7)(2,6,5,8)(15,24)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378194);