#2739034

order	512
nilpotency class	2
isoclinism	isoclinic class of #2735949
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×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2426 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,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, 2646230);