#8993807

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	#43193	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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)(17,18,23,26)(19,27,29,24)(20,30,31,21)(22,25,28,32)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,13)(15,16)(17,19,23,29)(18,24,26,27)(20,25,31,32)(21,28,30,22)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(2,8)(3,11)(4,13)(5,15)(20,31)(21,30)(22,28)(25,32)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,13)(15,16)(17,20,23,31)(18,21,26,30)(19,25,29,32)(22,24,28,27)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,4)(2,9)(3,12)(5,14)(6,13)(7,15)(8,16)(10,11)(17,21)(18,20)(19,28)(22,29)(23,30)(24,25)(26,31)(27,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,5,6,15)(2,10,8,12)(3,9,11,16)(4,14,13,7)(17,22,23,28)(18,25,26,32)(19,21,29,30)(20,24,31,27)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=ghi,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, 8901003);