#8942364

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

4239 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)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(33,34)(35,41)(36,37)(38,46)(39,40)(42,47)(43,45)(44,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,16)(13,14)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(34,40)(37,41)(42,46)(45,48);(2,6)(3,5)(9,12)(11,14)(18,23)(20,27)(21,29)(26,31)(33,35,39,36)(34,37,40,41)(38,44,47,43)(42,48,46,45);(1,4,7,13)(2,9,6,12)(3,11,5,14)(8,15,10,16)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,31)(30,32)(33,36,39,35)(34,41,40,37)(38,44,47,43)(42,48,46,45);(1,5,7,3)(2,10,6,8)(4,11,13,14)(9,15,12,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,37,39,41)(34,36,40,35)(38,45,47,48)(42,44,46,43);(1,6,7,2)(3,10,5,8)(4,12,13,9)(11,16,14,15)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8849560);