#385025

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi

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, 292221);