#8942373

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

4256 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)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32);(2,6)(7,9)(8,12)(14,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,32)(29,30)(33,34,38,39)(35,36,40,43)(37,41,45,46)(42,44,47,48);(2,6)(4,13)(7,9)(10,16)(18,22)(19,21)(25,28)(27,30)(34,39)(36,43)(37,45)(42,47);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,23,29)(18,25,22,28)(19,27,21,30)(24,31,26,32)(33,35)(34,36)(37,42)(38,40)(39,43)(41,44)(45,47)(46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,21,23,19)(18,26,22,24)(20,27,29,30)(25,31,28,32)(33,36)(34,40)(35,39)(37,44)(38,43)(41,47)(42,46)(45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22,23,18)(19,26,21,24)(20,28,29,25)(27,32,30,31)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|b^2=ghi,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, 8849569);