#198864

order	512
nilpotency class	3
isoclinism	isoclinic class of #198789
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6,8,11)(3,9,10,12)(17,18,22,27,23,28,21,26)(19,29,31,39,33,41,32,24)(20,34,35,42,37,25,36,40)(30,43,46,38,45,48,44,47);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19,23,33)(18,24)(20,30,37,45)(21,32,22,31)(25,38)(26,41)(27,29)(28,39)(34,47)(35,46,36,44)(40,48)(42,43);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,40)(27,42)(28,34)(29,43)(31,44)(32,46)(33,45)(39,47)(41,48);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,41)(25,40,34,42)(30,44,45,46)(38,43,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,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, 106060);