#5968591

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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1952 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)(9,10,13,14)(11,16,15,12)(17,18,21,24)(19,22,23,20)(26,28)(27,33)(29,37)(38,40);(2,5)(6,8)(17,19)(18,22)(20,24)(21,23)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(10,14)(11,15)(17,20,21,22)(18,23,24,19)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,20)(18,19)(21,22)(23,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,20)(21,23)(22,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=h,d^2=gh,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, 5875787);