#8406498

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

3997 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 8313694);