#382290

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, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,18)(17,19)(21,22,25,27,24,23,26,28);(1,2,3,4)(6,9)(10,12)(22,23)(27,28);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,16)(14,18,19,20);(7,11)(10,12)(13,16,17,15)(14,18,19,20)(21,23,25,28,24,22,26,27)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=fh,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,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, 289486);