#423168

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(6,9)(10,12)(13,14,18,24)(15,27,17,21)(16,30,33,42)(19,26,20,25)(22,35,41,34)(23,29,38,36)(28,40,32,39)(31,43,44,37);(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,31)(17,18)(19,36)(20,29)(22,39)(23,25)(24,27)(26,38)(28,34)(30,37)(32,35)(33,44)(40,41)(42,43);(1,2,3,4)(5,7,8,11)(6,10,9,12)(13,16,20,35,18,33,19,34)(14,22,26,42,24,41,25,30)(15,28,36,44,17,32,29,31)(21,37,23,40,27,43,38,39);(6,9)(10,12)(13,17,18,15)(14,23,24,38)(16,32,33,28)(19,36,20,29)(21,26,27,25)(22,40,41,39)(30,37,42,43)(31,34,44,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330364);