#9018076

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

4276 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,f^2=i,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, 8925272);