#695463

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

3932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(6,8)(11,15)(13,16)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28);(2,4)(6,8)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(21,30)(27,29);(1,2)(3,4)(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,29)(21,28)(22,26)(23,31)(25,27)(30,32);(9,11)(10,13)(12,15)(14,16)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28);(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^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, 602659);