#913728

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1831 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=fgi,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 820924);