#8943866

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851062);