#224930

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=ehi,d^2=e,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132126);