#464752

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1510 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,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, 371948);