#8831162

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

4281 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8738358);