#8942666

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

4056 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(33,34,39,41)(35,42,37,36)(38,40,47,48)(43,46,45,44);(2,6)(7,9)(8,12)(14,15)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(33,35,39,37)(34,36,41,42)(38,43,47,45)(40,44,48,46);(2,6)(4,13)(7,9)(10,16)(18,23)(20,27)(21,29)(26,31)(33,36,39,42)(34,35,41,37)(38,46,47,44)(40,45,48,43);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,31)(30,32)(33,34)(35,36)(37,42)(38,40)(39,41)(43,44)(45,46)(47,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,37,39,35)(34,36,41,42)(38,45,47,43)(40,44,48,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38,39,47)(34,40,41,48)(35,43,37,45)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,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, 8849862);