#382768

order	512
nilpotency class	3
isoclinism	isoclinic class of #188766
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1150 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)(12,15)(13,16)(14,19)(18,23)(20,24)(21,22);(2,5)(6,8)(9,10)(11,18,19,24)(12,15)(13,16)(14,20,17,23)(21,22);(3,7)(6,8)(9,11)(10,14,16,17)(12,20)(13,19)(15,18,22,24)(21,23);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289964);