#689777

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3110 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)(11,15)(13,16)(17,18,22,23)(19,25,26,20)(21,24,29,30)(27,31,32,28);(6,8)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(21,29)(28,31);(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,26)(18,20,23,25)(21,27,29,32)(24,28,30,31);(1,2)(3,4)(9,11,12,15)(10,13,14,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32);(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=g,d^2=gh,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, 596973);