#687719

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3722 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)(9,10,11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28);(6,8)(9,10,11,12)(14,19)(15,21)(17,25)(23,28);(5,6,7,8)(9,10)(11,12)(13,15,18,21)(14,16,19,24)(17,22,25,27)(20,23,26,28);(1,2)(3,4)(13,16)(14,15)(17,23)(18,24)(19,21)(20,22)(25,28)(26,27);(13,17,18,25)(14,20,19,26)(15,22,21,27)(16,23,24,28)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 594915);