#381137

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1175 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,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, 288333);