#8994007

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	#55149	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,18,22,21)(19,23,28,26)(20,24,29,30)(25,31,27,32)(34,39)(35,43)(36,46)(42,47);(1,3,6,9)(2,7,5,11)(4,10,12,13)(8,15,14,16)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,34)(35,40)(36,41)(37,47)(38,39)(42,44)(43,45)(46,48);(2,5)(3,9)(4,12)(15,16)(17,20,22,29)(18,24,21,30)(19,25,28,27)(23,31,26,32)(33,35,38,43)(34,40,39,45)(36,37,46,44)(41,47,48,42);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,13)(11,16)(17,21,22,18)(19,26,28,23)(20,24,29,30)(25,31,27,32)(33,36,38,46)(34,41,39,48)(35,44,43,37)(40,42,45,47);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,12)(14,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,32)(25,28)(26,31)(33,37,38,44)(34,42,39,47)(35,36,43,46)(40,48,45,41);(1,5,6,2)(3,11,9,7)(4,14,12,8)(10,16,13,15)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,34)(35,45)(36,41)(37,42)(38,39)(40,43)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=hi,e^2=g,f^2=gh,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, 8901203);