#425089

order	512
nilpotency class	3
isoclinism	isoclinic class of #419895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

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, 332285);