#416072

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=e,d^2=h,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,d^c=di,f^c=fi,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, 323268);