#190726

order	512
nilpotency class	3
isoclinism	isoclinic class of #160780
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=e,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,f^c=fi,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97922);