#1118237

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1856 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=f,d^2=gh,e^2=f,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, 1025433);