#5880893

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1869 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)(3,8)(4,7)(5,6)(9,10,12,14)(11,15,16,13)(17,18,21,20)(19,23,24,22)(26,30)(27,33)(28,36)(38,40);(2,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,24)(18,22,20,23)(26,30)(28,36)(31,34)(35,39);(1,3,6,4)(2,7,5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,4,6,3)(2,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,20,21,18)(19,23,24,22)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,5,6,2)(3,8,4,7)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,22)(21,24)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=fgh,c^2=fhi,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5788089);