#706463

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,6)(3,8)(10,13)(14,16)(18,22)(23,24)(25,26,31,33)(27,34,28,30)(29,32,38,40)(35,39,36,37);(18,22)(23,24)(25,27,31,28)(26,30,33,34)(29,35,38,36)(32,37,40,39);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,28,31,27)(26,30,33,34)(29,35,38,36)(32,39,40,37);(1,3)(2,7)(4,6)(5,8)(17,19,21,20)(18,23,22,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,40)(34,39);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)(25,30)(26,27)(28,33)(29,37)(31,34)(32,35)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 613659);