#5934149

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1458 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fghi,c^2=fi,d^2=fghi,e^2=f,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, 5841345);