#688027

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

4626 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=h,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, 595223);