#727009

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1714 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=hi,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, 634205);