#8876374

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	#50015	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4132 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=h,e^2=h,f^2=i,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, 8783570);