#8977154

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	#48071	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(34,36)(39,43)(40,44)(47,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(33,34,38,36)(35,39,42,43)(37,40,46,44)(41,47,45,48);(2,6)(3,10)(4,13)(14,15)(18,23)(19,27)(20,30)(21,31)(33,35,38,42)(34,39,36,43)(37,45,46,41)(40,48,44,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,47)(44,48)(45,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,37,38,46)(34,40,36,44)(35,41,42,45)(39,47,43,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=g,e^2=h,f^2=g,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, 8884350);