#187111

order	512
nilpotency class	3
isoclinism	isoclinic class of #186967
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 94307);