#187116

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	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,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, 94312);