#4973849

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

1438 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)(9,10,14,12)(11,16,13,15)(18,21)(22,23)(25,26)(27,33)(28,32)(29,35)(30,31)(34,39)(36,40)(37,38);(17,18,20,21)(19,23,24,22)(26,31)(29,33)(32,38)(36,39);(2,6)(3,8)(9,11,14,13)(10,15,12,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,29)(28,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,3,5,8)(2,7,6,4)(9,12,14,10)(11,15,13,16)(25,28,30,37)(26,32,31,38)(27,34,35,40)(29,36,33,39);(1,4,5,7)(2,8,6,3)(9,13,14,11)(10,15,12,16)(17,19,20,24)(18,22,21,23)(25,29)(26,27)(28,36)(30,33)(31,35)(32,34)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=fgi,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, 4881045);