#187864

order	512
nilpotency class	3
isoclinism	isoclinic class of #8528
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	#62759	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2286 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:

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

Polycyclic presentation:

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

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, 95060);