#8943837

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

4143 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,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, 8851033);