#4730144

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1474 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)(3,8)(4,7)(5,6)(9,10)(11,12)(13,16)(14,15)(20,24)(22,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2,6,5)(3,7,4,8)(9,11)(10,14)(12,16)(13,15)(17,18,21,19)(20,22,24,23)(26,27)(31,33)(32,34)(38,39);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,15)(14,16)(17,19,21,18)(20,23,24,22)(25,27,30,26)(28,33,36,31)(29,32,37,34)(35,38,40,39);(1,4,6,3)(2,7,5,8)(17,20,21,24)(18,22,19,23)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,5,6,2)(3,7,4,8)(9,12)(10,15)(11,16)(13,14)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=ghi,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, 4637340);