#5880922

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

1925 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:

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

Polycyclic presentation:

2^9|a^2=fghi,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, 5788118);