#380772

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1466 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287968);