#522091

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=fg,e^2=f,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, 429287);