#198212

order	512
nilpotency class	3
isoclinism	isoclinic class of #198135
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	C₂×C₄²	C₂⁴
prime radical	C₂×#3219	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1075 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,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, 105408);