#5880866

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1630 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fh,c^2=hi,d^2=fgi,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, 5788062);