#199195

order	512
nilpotency class	3
isoclinism	isoclinic class of #199194
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	#60805	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106391);