#8873585

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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)(3,9)(4,11)(5,13)(6,7)(8,15)(10,12)(14,16)(17,18)(19,24)(20,30)(21,26)(22,23)(25,29)(27,32)(28,31)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(1,3,6,11)(2,4,7,9)(5,14,15,10)(8,16,13,12)(18,23)(21,30)(24,25)(28,32)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(2,7)(4,9)(8,13)(12,16)(17,19,22,29)(18,24,23,25)(20,27,26,31)(21,28,30,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,4,6,9)(2,3,7,11)(5,12,15,16)(8,10,13,14)(17,19)(18,25)(20,31)(21,28)(22,29)(23,24)(26,27)(30,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,20,22,26)(18,21,23,30)(19,27,29,31)(24,28,25,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,5,6,15)(2,8,7,13)(3,10,11,14)(4,12,9,16)(17,21)(18,26)(19,28)(20,23)(22,30)(24,31)(25,27)(29,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=hi,e^2=ghi,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, 8780781);