#509208

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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

761 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416404);