#1209149

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=fh,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1116345);