#3827211

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1714 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=fgi,d^2=fgi,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3734407);