#381333

order	512
nilpotency class	3
isoclinism	isoclinic class of #214664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1265 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=eg,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,d^c=de,f^c=fi,f^d=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, 288529);