#8979749

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

4211 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8886945);