#4198858

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
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

1742 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ghi,c^2=fgh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4106054);