#8993932

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

4237 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)(18,23)(19,25)(24,30)(26,31)(33,34)(35,41)(36,44)(37,40)(38,43)(39,46)(42,47)(45,48);(1,3,6,11)(2,7,8,16)(4,10,13,14)(5,12,15,9)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(2,8)(3,11)(4,13)(5,15)(19,25)(20,27)(21,29)(24,30)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,4)(2,9)(3,10)(5,7)(6,13)(8,12)(11,14)(15,16)(17,19,22,25)(18,20,23,27)(21,26,29,31)(24,28,30,32)(33,35)(34,38)(36,42)(37,43)(39,47)(40,41)(44,48)(45,46);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(1,5,6,15)(2,10,8,14)(3,12,11,9)(4,7,13,16)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,28,27,32)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=ghi,f^2=gh,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, 8901128);