#198810

order	512
nilpotency class	3
isoclinism	isoclinic class of #198807
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	#60801	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1081 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:

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

Polycyclic presentation:

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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106006);