#678225

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=f,d^2=fgi,e^2=fg,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, 585421);