#8406974

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

4312 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)(3,8)(4,5)(6,7)(9,13)(10,16)(11,14)(12,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,39,43)(35,40,45,38)(36,42,37,41)(44,48,46,47);(2,7)(4,13)(8,15)(10,14)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(33,35,39,45)(34,40,43,38)(36,46,37,44)(41,48,42,47);(2,7)(5,9)(8,15)(11,16)(19,28)(22,31)(24,32)(26,29)(33,36,39,37)(34,41,43,42)(35,44,45,46)(38,48,40,47);(1,3,6,12)(2,8,7,15)(4,14,13,10)(5,11,9,16)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,37,39,36)(34,42,43,41)(35,44,45,46)(38,48,40,47);(1,4)(2,9)(3,10)(5,7)(6,13)(8,11)(12,14)(15,16)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,35)(34,38)(36,44)(37,46)(39,45)(40,43)(41,48)(42,47);(1,5,6,9)(2,4,7,13)(3,11,12,16)(8,14,15,10)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,38)(34,35)(36,47)(37,48)(39,40)(41,46)(42,44)(43,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314170);