#404715

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1225 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)(13,14)(15,25)(16,30)(17,24)(18,21)(19,23)(20,27)(22,31)(26,40)(28,36)(29,34)(32,38)(33,37)(35,42)(39,41)(43,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,27,18,28,19,29)(14,20,23,25,21,34,24,36)(16,26,31,41,32,42,33,43)(22,35,30,39,37,40,38,44);(1,2,3,4)(6,9)(10,12)(14,21)(20,34)(22,37)(23,24)(25,36)(30,38)(35,40)(39,44);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,31)(18,32)(19,33)(20,35)(21,37)(23,30)(24,38)(25,39)(27,41)(28,42)(29,43)(34,40)(36,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311911);