#368321

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=gi,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi

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, 275517);