#8977336

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

4235 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,e^2=h,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, 8884532);