#4967483

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

1442 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)(3,4)(5,6)(7,8)(11,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(2,6)(4,7)(17,19,22,20)(18,23,21,24)(27,33)(31,34)(35,39)(38,40);(1,2,5,6)(3,7,8,4)(9,10,12,14)(11,13,15,16)(17,20,22,19)(18,23,21,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,15)(14,16)(17,21,22,18)(19,24,20,23)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,4,5,7)(2,3,6,8)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fhi,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, 4874679);