#8942526

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

4037 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 8849722);