#2722210

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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

1650 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2629406);