#5880890

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

1865 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fgh,c^2=hi,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, 5788086);