#8844495

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

4180 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)(3,7,9,4)(5,13,15,10)(11,12,14,16)(17,18,23,26)(19,24,20,25)(21,31,32,22)(27,28,29,30)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,47)(46,48);(1,3)(2,7)(4,8)(5,14)(6,9)(10,12)(11,15)(13,16)(17,19,23,20)(18,24,26,25)(21,27,32,29)(22,30,31,28)(34,37)(36,45)(39,41)(42,46);(2,8)(3,9)(10,13)(11,14)(19,20)(24,25)(27,29)(28,30)(33,35,38,43)(34,39,37,41)(36,42,45,46)(40,47,44,48);(1,3)(2,7)(4,8)(5,11)(6,9)(10,16)(12,13)(14,15)(17,20,23,19)(18,25,26,24)(21,27,32,29)(22,30,31,28)(33,35)(34,39)(36,46)(37,41)(38,43)(40,48)(42,45)(44,47);(1,4,6,7)(2,9,8,3)(5,12,15,16)(10,14,13,11)(17,21)(18,22)(19,27)(20,29)(23,32)(24,30)(25,28)(26,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,47)(41,48)(43,46);(1,5)(2,10)(3,11)(4,12)(6,15)(7,16)(8,13)(9,14)(17,22)(18,21)(19,28)(20,30)(23,31)(24,29)(25,27)(26,32)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,48,46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8751691);