#187107

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3216 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)(11,20)(12,24,26,17)(13,18)(14,27)(15,19)(16,22)(21,35,38,40)(23,31)(25,34,29,33)(28,32)(30,37,39,36);(2,5)(3,7)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,30)(18,31)(19,32)(20,27)(24,39)(25,36)(26,38)(29,37)(33,40)(34,35);(2,5)(3,7)(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,33)(19,24)(20,35)(22,36)(23,38)(27,34)(28,37)(31,40)(32,39);(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,25,26,29)(16,31,32,20)(17,33,24,34)(21,36,38,37)(30,40,39,35)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,g^2=hi,b^a=be,c^a=cf,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94303);