#4955293

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

1434 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fhi,c^2=gh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4862489);