#2889750

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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

1538 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,19)(20,23,24,22)(25,26)(27,30)(28,36)(29,33)(31,35)(32,37)(34,39)(38,40);(2,6)(7,8)(10,13)(14,15)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,4,5,3)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20)(18,22)(19,23)(21,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)

Polycyclic presentation:

2^9|a^2=fghi,b^2=g,c^2=i,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

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, 2796946);