#521520

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428716);