#693114

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2848 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)(9,10,13,16)(11,12,15,14)(17,18,21,25)(19,23,27,22)(20,24,29,32)(26,31,28,30);(6,8)(17,19,21,27)(18,22,25,23)(20,28,29,26)(24,31,32,30);(5,6,7,8)(9,11,13,15)(10,14,16,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,29)(30,32);(1,2)(3,4)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,31)(29,32);(9,12,13,14)(10,15,16,11)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=g,e^2=fgh,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, 600310);