#4427699

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1438 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fg,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4334895);