#456167

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1120 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:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,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, 363363);