#197568

order	512
nilpotency class	3
isoclinism	isoclinic class of #197354
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1163 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=h,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dhi,e^c=eh,g^c=gh

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, 104764);