#1183614

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

1510 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fhi,d^2=f,e^2=fgh,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, 1090810);