#8993912

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

4223 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=hi,e^2=g,f^2=h,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, 8901108);