#381670

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

1082 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:

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

Polycyclic presentation:

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

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, 288866);