#8550894

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458090);