#1241789

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

1550 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)(6,8)(11,12)(13,15)(17,18,21,20)(19,22,23,24)(25,26,30,31)(27,32,33,28)(29,34,37,38)(35,39,40,36);(9,10,14,16)(11,15,12,13)(19,23)(22,24)(26,31)(27,33)(29,37)(36,39);(9,11,14,12)(10,15,16,13)(17,19,21,23)(18,22,20,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,13,16,15)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,3,4,7)(2,6,5,8)(9,13)(10,11)(12,16)(14,15)(17,20,21,18)(19,24,23,22)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fgh,d^2=fg,e^2=ghi,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, 1148985);