#386820

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fg,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=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, 294016);