#8976963

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

4060 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(33,34,37,41)(35,40,42,38)(36,39,45,48)(43,47,44,46);(2,6)(7,9)(8,12)(14,15)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(35,42)(38,40)(43,44)(46,47);(2,6)(3,10)(4,13)(14,15)(18,23)(19,27)(20,30)(21,31)(33,35,37,42)(34,38,41,40)(36,44,45,43)(39,47,48,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,36,37,45)(34,39,41,48)(35,43,42,44)(38,46,40,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,35)(34,40)(36,44)(37,42)(38,41)(39,46)(43,45)(47,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,f^2=g,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, 8884159);