#466814

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 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)(5,6,8,9)(7,10,11,12)(13,14)(15,20)(16,18)(17,30)(19,22)(21,33)(23,34)(24,27)(25,39)(26,28)(29,32)(31,38)(35,37)(36,42)(40,44)(41,43);(6,9)(10,12)(13,15,19,27,18,26,34,35)(14,20,22,24,16,28,23,37)(17,25,32,41,31,40,33,42)(21,36,30,39,29,43,38,44);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16,18,14)(15,24,26,37)(17,29,31,21)(19,22,34,23)(20,35,28,27)(25,39,40,44)(30,33,38,32)(36,42,43,41);(1,3,4,2)(7,11)(10,12)(13,17)(14,21)(15,25)(16,29)(18,31)(19,32)(20,36)(22,30)(23,38)(24,39)(26,40)(27,41)(28,43)(33,34)(35,42)(37,44)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=ef,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,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, 374010);