#2736111

order	512
nilpotency class	2
isoclinism	isoclinic class of #2735949
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2256 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=fg,d^2=fg,e^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, 2643307);