#9030075

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

4146 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)(3,8,5,11)(4,10,15,14)(6,13,16,12)(17,18,23,26)(19,27,29,24)(20,30,31,21)(22,25,28,32)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(1,3)(2,8)(4,13)(5,7)(6,14)(9,11)(10,16)(12,15)(17,19,23,29)(18,24,26,27)(20,25,31,32)(21,28,30,22)(34,36)(39,41)(40,44)(47,48);(2,9)(3,5)(4,15)(6,16)(20,31)(21,30)(22,28)(25,32)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,20,23,31)(18,21,26,30)(19,25,29,32)(22,24,28,27)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,21)(18,20)(19,28)(22,29)(23,30)(24,25)(26,31)(27,32)(33,36,38,34)(35,39,43,41)(37,44,46,40)(42,47,45,48);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,22,23,28)(18,25,26,32)(19,21,29,30)(20,24,31,27)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8937271);