#5078911

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1720 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4986107);