#5073703

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1448 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)(9,10,15,16)(11,12,13,14)(17,18)(19,20)(21,23)(22,24)(25,26,30,28)(27,31,33,35)(29,36,38,32)(34,40,37,39);(2,6)(3,8)(9,11,15,13)(10,12,16,14)(17,19)(18,22)(20,23)(21,24)(26,28)(27,33)(32,36)(34,37);(3,8)(4,7)(9,12,15,14)(10,11,16,13)(17,18,21,23)(19,22,24,20)(25,27)(26,31)(28,35)(29,37)(30,33)(32,40)(34,38)(36,39);(1,3,5,8)(2,7,6,4)(9,13,15,11)(10,12,16,14)(17,20)(18,19)(21,22)(23,24)(25,28,30,26)(27,31,33,35)(29,36,38,32)(34,39,37,40);(1,4,5,7)(2,8,6,3)(9,14,15,12)(10,11,16,13)(17,18)(19,20)(21,23)(22,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4980899);