#439569

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=efi,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi

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, 346765);