#409066

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1183 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)(8,9)(11,12)(13,14)(15,21)(16,30)(17,24)(18,23)(19,25)(20,27)(22,33)(26,35)(28,37)(29,36)(31,39)(32,41)(34,40)(38,43)(42,44);(1,2,4,3)(6,9)(10,12)(13,15,18,28,19,29,17,27)(14,20,24,36,25,37,23,21)(16,31,33,42,34,26,32,43)(22,39,30,38,41,35,40,44);(1,3,4,2)(13,15,19,29)(14,21,25,36)(16,26,34,31)(17,27,18,28)(20,23,37,24)(22,38,41,44)(30,35,40,39)(32,42,33,43);(5,7,8,11)(6,10,9,12)(13,16,17,32,19,34,18,33)(14,22,23,40,25,41,24,30)(15,26,27,42,29,31,28,43)(20,35,21,38,37,39,36,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316262);