#8979152

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

4067 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,e^2=gh,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, 8886348);