#765914

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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

1620 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:

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

Polycyclic presentation:

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

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, 673110);