#8979583

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

4237 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,23,21)(19,20,27,24)(22,29,30,25)(26,31,32,28)(33,34,39,42)(35,43,45,40)(36,46,47,37)(38,41,44,48);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19,23,27)(18,24,21,20)(22,26,30,32)(25,31,29,28)(33,35,39,45)(34,40,42,43)(36,41,47,48)(37,44,46,38);(2,6)(3,4)(5,13)(15,16)(18,21)(20,24)(22,30)(26,32)(36,47)(37,46)(38,44)(41,48);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,20)(18,19)(21,27)(22,28)(23,24)(25,26)(29,32)(30,31)(33,36,39,47)(34,37,42,46)(35,41,45,48)(38,40,44,43);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,21,23,18)(19,24,27,20)(22,29,30,25)(26,31,32,28)(33,37)(34,36)(35,44)(38,45)(39,46)(40,41)(42,47)(43,48);(1,6,7,2)(3,9,4,8)(5,14,13,10)(11,16,12,15)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)(33,38,39,44)(34,41,42,48)(35,37,45,46)(36,40,47,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gh,e^2=h,f^2=gh,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, 8886779);