#1687406

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=ghi,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1594602);