#5118132

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1772 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=gh,d^2=gi,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, 5025328);