#3563578

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1555 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=i,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3470774);