#5119595

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1516 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)(4,6)(7,8)(9,10,14,12)(11,15,13,16)(19,20)(23,24)(25,26)(27,31)(28,30)(29,36)(32,38)(33,35)(34,40)(37,39);(1,2,4,6)(3,7,8,5)(9,11,14,13)(10,15,12,16)(18,22)(23,24)(26,28)(27,33)(32,36)(34,37);(1,2,4,6)(3,5,8,7)(9,12,14,10)(11,15,13,16)(17,18,21,22)(19,23,20,24)(25,27,30,33)(26,31,28,35)(29,37,38,34)(32,40,36,39);(1,3)(2,5)(4,8)(6,7)(17,19,21,20)(18,23,22,24)(25,28,30,26)(27,31,33,35)(29,36,38,32)(34,39,37,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,11)(10,15,12,16)(17,20,21,19)(18,24,22,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=i,e^2=fgh,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, 5026791);