#4245084

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1811 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)(3,7)(4,5)(6,8)(9,10,13,14)(11,12,16,15)(17,18,20,23)(19,21,24,22)(25,26,30,27)(28,31,36,33)(29,32,38,34)(35,39,37,40);(3,8)(6,7)(9,11,13,16)(10,12,14,15)(17,19)(18,21)(20,24)(22,23)(26,27)(31,33)(32,34)(39,40);(2,5)(6,7)(10,14)(12,15)(17,19,20,24)(18,22,23,21)(25,27,30,26)(28,33,36,31)(29,32,38,34)(35,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,15)(10,11,14,16)(17,19)(18,22)(20,24)(21,23)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,40)(36,37)

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,c^2=h,d^2=hi,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, 4152280);