#8843458

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

4130 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8750654);