#5729167

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1562 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=fgh,e^2=g,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, 5636363);