#5988616

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5895812);