#458206

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

948 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)(7,10)(8,12)(9,11)(13,14)(15,20)(16,21)(17,34)(18,23)(19,25)(22,32)(24,35)(26,36)(27,43)(28,37)(29,41)(30,38)(31,39)(33,40)(42,44);(1,2,4,3)(5,7,9,8)(6,10,11,12)(13,15,19,30)(14,20,25,38)(16,31,17,27)(18,29,35,28)(21,39,34,43)(22,36,40,44)(23,41,24,37)(26,33,42,32);(2,3)(5,8,9,7)(6,10,11,12)(13,16,19,17)(14,21,25,34)(15,26,30,42)(18,33,35,32)(20,36,38,44)(22,23,40,24)(27,29,31,28)(37,43,41,39);(1,3,4,2)(13,17,19,16)(14,22,25,40)(15,27,30,31)(18,33,35,32)(20,36,38,44)(21,23,34,24)(26,29,42,28)(37,39,41,43)

Polycyclic presentation:

2^9|b^2=efi,c^2=fi,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, 365402);