#4757506

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, 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=g,d^2=h,e^2=ghi,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, 4664702);