#8942603

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

4096 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=h,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, 8849799);