#186383

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62815	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2251 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93579);