#509210

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:

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

Polycyclic presentation:

2^9|a^2=e,b^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, 416406);