#522678

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3816 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^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, 429874);