#714403

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

1680 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=i,d^2=fh,e^2=fg,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, 621599);