#502146

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fh,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409342);