#4985202

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

1426 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)(3,4,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,20)(19,23,21,24)(25,26)(27,33)(28,34)(29,32)(30,31)(35,39)(36,40)(37,38);(9,10)(11,13)(12,14)(15,16)(17,19,22,21)(18,23,20,24)(26,31)(27,34)(29,37)(36,39);(2,6)(3,8)(9,10,12,14)(11,13,16,15)(17,19,22,21)(18,24,20,23)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,40)(38,39);(1,3,5,8)(2,7,6,4)(9,10)(11,13)(12,14)(15,16)(17,20,22,18)(19,24,21,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,4,5,7)(2,8,6,3)(9,11)(10,13)(12,16)(14,15)(17,21,22,19)(18,24,20,23)(25,29,30,37)(26,32,31,38)(27,35,34,40)(28,36,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=fghi,c^2=gi,d^2=gh,e^2=fgh,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, 4892398);