#4928681

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=ghi,c^2=hi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=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, 4835877);