#382983

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1594 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,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, 290179);