#678055

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

1784 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,16,12,13)(17,18,22,24)(19,20,21,23)(25,26,28,29)(27,30,31,32);(2,5)(6,7)(9,11,14,12)(10,13,15,16)(17,19,22,21)(18,20,24,23)(26,29)(27,31);(1,3)(2,6)(4,8)(5,7)(9,10,14,15)(11,16,12,13)(17,20,22,23)(18,19,24,21)(25,26)(27,30)(28,29)(31,32);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,13,15,16)(17,18)(19,20)(21,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,13)(10,11)(12,15)(14,16)(17,21,22,19)(18,23,24,20)(25,27,28,31)(26,30,29,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fg,d^2=ghi,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, 585251);