#4664067

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1378 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)(9,10,12,14)(11,15,16,13)(20,24)(22,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(2,7)(3,4)(9,11)(10,13)(12,16)(14,15)(17,18,21,19)(20,22,24,23)(26,30)(31,37)(32,35)(38,39);(1,3,6,4)(2,5,7,8)(9,10,12,14)(11,13,16,15)(17,19,21,18)(20,23,24,22)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(1,4,6,3)(2,5,7,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,22)(19,23)(21,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(1,5)(2,3)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(25,28,29,36)(26,32,30,35)(27,33,34,40)(31,38,37,39)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=gi,d^2=h,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,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, 4571263);