#509852

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

817 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:

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

Polycyclic presentation:

2^9|a^2=ef,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,h^b=hi,f^c=fi,e^d=eh,f^d=fi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417048);