#416070

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1170 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)(5,6)(7,11)(8,12)(9,10)(13,14)(15,25)(16,29)(17,24)(18,23)(19,22)(20,27)(21,32)(26,41)(28,39)(30,44)(31,38)(33,36)(34,40)(35,42)(37,43);(6,10)(11,12)(13,15,18,33,19,28,17,27)(14,20,24,39,22,36,23,25)(16,30,34,43,32,42,31,26)(21,37,40,44,29,41,38,35);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,31,19,32,18,34)(14,21,23,38,22,29,24,40)(15,26,27,42,28,43,33,30)(20,35,25,41,36,44,39,37);(5,8,9,7)(6,11,10,12)(14,22)(20,36)(21,29)(23,24)(25,39)(35,44)(37,41)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323266);