#2788802

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=f,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,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, 2695998);