#616896

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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

1858 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fg,d^2=fh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 524092);