#826402

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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

1732 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=fgh,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 733598);