#8850576

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

4028 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=ghi,e^2=g,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, 8757772);