#5203554

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1786 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=gi,d^2=gh,e^2=i,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, 5110750);