#837457

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

1706 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)(9,10,14,16)(11,13,12,15)(17,18,22,21)(19,24,20,23);(9,11,14,12)(10,15,16,13)(17,18,22,21)(19,23,20,24)(25,26,28,30)(27,29,31,32);(17,19,22,20)(18,23,21,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,13,16,15)(17,20,22,19)(18,24,21,23);(1,3,4,7)(2,6,5,8)(9,13)(10,12)(11,16)(14,15)(17,21,22,18)(19,24,20,23)(25,27,28,31)(26,29,30,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=g,d^2=fg,e^2=fhi,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, 744653);