#395365

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1380 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)(5,6,9,11)(7,12,8,10)(13,14)(15,20)(16,29)(17,31)(18,22)(19,23)(21,36)(24,25)(26,40)(27,38)(28,39)(30,43)(32,33)(34,35)(37,41)(42,44);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,30)(17,25)(18,26)(19,28)(21,38)(22,40)(23,39)(24,31)(27,36)(29,43)(32,41)(33,37)(34,44)(35,42);(2,4)(13,16,19,32)(14,17,23,34)(15,24,28,42)(18,35,36,31)(20,37,39,43)(21,33,22,29)(25,38,44,40)(26,30,27,41);(5,8,9,7)(6,10,11,12)(13,17,19,34)(14,16,23,32)(15,25,28,44)(18,33,36,29)(20,30,39,41)(21,35,22,31)(24,38,42,40)(26,37,27,43)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 302561);