#674117

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

1804 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)(3,8)(4,6)(5,7)(10,14)(11,16)(17,18)(19,21)(20,22)(23,24)(25,26,29,28)(27,30,31,32);(1,3,5,4)(2,6,7,8)(18,19)(20,24)(27,31)(30,32);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,18)(20,23,24,22)(25,27)(26,30)(28,32)(29,31);(1,4,5,3)(2,6,7,8)(9,11,13,16)(10,15,14,12)(25,26,29,28)(27,30,31,32);(1,2)(3,8)(4,6)(5,7)(9,12,13,15)(10,16,14,11)(17,20,21,24)(18,22,19,23)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=gi,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, 581313);