#8534763

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	#37493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4210 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

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, 8441959);