#8993910

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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)(17,18,22,21)(19,23,28,26)(20,24,29,30)(25,31,27,32)(33,34,39,37)(35,42,41,38)(36,44,46,40)(43,45,47,48);(1,3,6,11)(2,7,8,16)(4,10,13,14)(5,12,15,9)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(34,37)(35,41)(40,44)(43,47);(2,8)(3,11)(4,13)(5,15)(17,20,22,29)(18,24,21,30)(19,25,28,27)(23,31,26,32)(33,35)(34,38)(36,43)(37,42)(39,41)(40,45)(44,48)(46,47);(1,4)(2,9)(3,10)(5,7)(6,13)(8,12)(11,14)(15,16)(17,21,22,18)(19,26,28,23)(20,24,29,30)(25,31,27,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13)(17,20)(18,24)(19,27)(21,30)(22,29)(23,32)(25,28)(26,31)(33,37,39,34)(35,42,41,38)(36,44,46,40)(43,48,47,45);(1,5,6,15)(2,10,8,14)(3,12,11,9)(4,7,13,16)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,38,39,42)(34,41,37,35)(36,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,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, 8901106);