#8633856

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

4199 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541052);