#8811192

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
prime radical	#48095	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8718388);