#4895494

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2286 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^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.

Relationships

maximal subgroup of	#72267607
maximal normal subgroup of	#72267607
maximal quotient of	#72267613

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4802690);