#719111

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,d^2=fi,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, 626307);