#2746383

order	512
nilpotency class	2
isoclinism	isoclinic class of #2735949
automorphism	Aut(G) order 2097152
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

1700 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)(6,8)(9,10,13,14)(11,15,16,12)(19,23)(21,24)(25,26)(27,30)(28,33)(29,34)(31,36)(32,38)(35,39)(37,40);(3,7)(6,8)(11,16)(12,15)(17,18,20,22)(19,21,23,24)(26,27)(31,33)(32,34)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,27,30,26)(28,31,36,33)(29,34,38,32)(35,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,19,20,23)(18,21,22,24)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,40)(36,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=f,d^2=i,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, 2653579);