#465181

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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

1105 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:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=fi,d^2=e,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, 372377);