#8321987

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	#45688	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18)(19,24)(20,26)(21,22)(23,25)(27,29)(28,31)(30,32);(2,7)(3,12)(9,14)(10,13)(17,19,23,22)(18,24,25,21)(20,28,30,27)(26,31,32,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(2,7)(5,16)(8,11)(9,14)(18,25)(19,22)(26,32)(27,28)(34,39)(37,47)(40,45)(41,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,23,30)(18,26,25,32)(19,27,22,28)(21,31,24,29)(33,35)(34,40)(36,42)(37,41)(38,44)(39,45)(43,46)(47,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21)(18,19)(20,29)(22,25)(23,24)(26,28)(27,32)(30,31)(33,36)(34,41)(35,43)(37,45)(38,46)(39,48)(40,47)(42,44);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,19)(18,21,25,24)(20,28,30,27)(26,31,32,29)(33,37,38,47)(34,42,39,43)(35,41,44,48)(36,45,46,40)

Polycyclic presentation:

2^9|b^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8229183);