#4895470

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2238 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)(3,6)(4,7)(5,8)(9,10,13,15)(11,12,16,14)(17,18)(19,20)(21,24)(22,23)(27,33)(28,36)(30,34)(31,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,26,29,32)(27,34,33,30)(28,31,36,38)(35,40,39,37);(17,20)(18,23)(19,24)(21,22)(25,27,29,33)(26,30,32,34)(28,35,36,39)(31,37,38,40);(1,3)(2,6)(4,8)(5,7)(9,12,13,14)(10,11,15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(1,2)(3,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,31)(27,35)(29,36)(30,37)(32,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=i,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	#72267545
maximal normal subgroup of	#72267545
maximal quotient of	#72267561, #72267564

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4802666);