#8993896

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

4158 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=hi,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, 8901092);