#8942552

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=gh,e^2=gh,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, 8849748);