#618707

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=f,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 525903);