#5123755

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=hi,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, 5030951);