#5203460

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1672 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)(3,7)(9,10,13,12)(11,14,16,15)(17,18,22,20)(19,23,21,24)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,37)(38,39);(1,2,5,4)(3,8,7,6)(9,10)(11,15)(12,13)(14,16)(17,19,22,21)(18,23,20,24)(26,28)(29,37)(31,33)(34,38);(2,4)(3,7)(9,11,13,16)(10,14,12,15)(17,20,22,18)(19,23,21,24)(25,27)(26,31)(28,33)(29,38)(30,35)(32,40)(34,37)(36,39);(1,3,5,7)(2,6,4,8)(9,12,13,10)(11,14,16,15)(17,19,22,21)(18,24,20,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(1,4,5,2)(3,8,7,6)(9,11)(10,15)(12,14)(13,16)(17,21,22,19)(18,23,20,24)(25,29,30,37)(26,32,28,36)(27,34,35,38)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fi,d^2=gh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5110656);