#4948681

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

1548 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=g,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, 4855877);