#190651

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

815 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)(11,15)(12,20,21,24)(13,18)(14,17)(16,19,23,22)(25,26,29,33,30,31,39,32)(27,28,35,37,34,38,40,36);(2,5)(6,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,20,23,24)(25,27,30,34)(26,28,31,38)(29,35,39,40)(32,36,33,37);(9,12)(10,16)(11,19)(13,22)(14,21)(15,20)(17,23)(18,24)(25,28,30,38)(26,27,31,34)(29,37,39,36)(32,40,33,35);(1,3,4,7)(2,6,5,8)(10,17)(15,18)(16,23)(20,24)(26,31)(27,34)(32,33)(35,40)

Polycyclic presentation:

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

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, 97847);