#1126162

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

1656 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,c^2=h,d^2=fg,e^2=h,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, 1033358);