#8993803

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	#40153	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4128 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=hi,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, 8900999);