#8943942

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

4075 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=gh,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, 8851138);