#381173

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^c=de,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288369);