#8943806

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=g,e^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, 8851002);