#2747079

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×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1656 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)(3,5)(4,7)(6,8)(10,12)(11,13)(17,18,22,24)(19,20,23,21)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,2,5,3)(4,6,8,7)(9,10,14,12)(11,15,13,16)(17,19)(18,21)(20,24)(22,23)(26,30)(27,33)(32,36)(35,39);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(17,20,22,21)(18,19,24,23)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,3,5,2)(4,6,8,7)(9,12,14,10)(11,15,13,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,16,12,15)(17,21,22,20)(18,23,24,19)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=i,e^2=fh,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, 2654275);