#198804

order	512
nilpotency class	3
isoclinism	isoclinic class of #198798
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

1080 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)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18)(19,24,31,40)(20,33)(21,28)(22,26)(23,27)(25,34)(29,43,45,48)(30,39,32,38)(35,42)(36,41)(37,46,47,44);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19,23,31)(18,24)(20,29,35,45)(21,30,22,32)(25,37)(26,38)(27,40)(28,39)(33,43)(34,44,36,46)(41,47)(42,48);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,33)(27,41)(28,42)(30,44)(31,45)(32,46)(38,43)(39,48)(40,47);(17,21,23,22)(18,26,27,28)(19,30,31,32)(20,34,35,36)(24,38,40,39)(25,33,41,42)(29,44,45,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,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, 106000);