#674136

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

1780 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)(3,4)(5,6)(7,8)(9,10,14,13)(11,16,15,12)(17,18,22,23)(19,24,20,21)(25,26)(27,29)(28,31)(30,32);(2,6)(4,7)(17,19,22,20)(18,21,23,24)(25,27,30,28)(26,31,32,29);(9,11)(10,12)(13,16)(14,15)(17,18,22,23)(19,24,20,21)(25,28,30,27)(26,31,32,29);(1,3)(2,7)(4,6)(5,8)(9,12,14,16)(10,15,13,11)(17,20,22,19)(18,21,23,24);(1,4,5,7)(2,3,6,8)(9,13,14,10)(11,16,15,12)(17,21)(18,19)(20,23)(22,24)(25,29,30,31)(26,27,32,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 581332);