#684020

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

1708 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)(3,5)(4,7)(6,8)(9,10,14,15)(11,16,13,12)(17,18,22,24)(19,20,21,23)(25,26)(27,31)(28,30)(29,32);(1,3,6,4)(2,7,8,5)(9,11,14,13)(10,12,15,16)(17,19,22,21)(18,20,24,23)(25,26,28,30)(27,29,32,31);(9,12)(10,11)(13,15)(14,16)(17,20,22,23)(18,19,24,21)(25,27)(26,29)(28,32)(30,31);(1,4,6,3)(2,7,8,5)(9,13,14,11)(10,12,15,16)(17,18)(19,20)(21,23)(22,24);(1,5,6,7)(2,3,8,4)(9,12)(10,11)(13,15)(14,16)(17,21,22,19)(18,23,24,20)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=fgi,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, 591216);