#719179

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

2646 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)(10,14)(11,16)(17,18,22,26)(19,27,30,23)(20,21,25,24)(28,29,31,32);(6,8)(17,19,22,30)(18,23,26,27)(20,28,25,31)(21,32,24,29);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,31)(21,29)(22,30)(24,32)(25,28)(26,27);(1,2,3,4)(9,11,13,16)(10,15,14,12)(17,20,22,25)(18,24,26,21)(19,28,30,31)(23,29,27,32);(9,12,13,15)(10,16,14,11)(17,21,22,24)(18,25,26,20)(19,29,30,32)(23,28,27,31)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,e^2=g,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, 626375);