#8256577

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

4076 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)(7,8)(10,14)(11,13)(15,16)(17,18,24,28)(19,29,31,25)(20,21,22,26)(23,27,30,32)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,47,45,44);(2,8)(5,15)(9,12)(10,13)(17,19)(18,25)(20,27)(21,23)(22,32)(24,31)(26,30)(28,29)(33,35,38,43)(34,39,41,36)(37,42,46,45)(40,47,48,44);(1,3,7,4)(2,9,8,12)(5,10,15,13)(6,14,16,11)(17,20,24,22)(18,26,28,21)(19,27,31,32)(23,25,30,29)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(1,4,7,3)(2,9,8,12)(5,13,15,10)(6,14,16,11)(17,21)(18,20)(19,30)(22,28)(23,31)(24,26)(25,32)(27,29);(1,5)(2,6)(3,10)(4,13)(7,15)(8,16)(9,14)(11,12)(17,22,24,20)(18,21,28,26)(19,27,31,32)(23,25,30,29)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,6)(2,5)(3,11)(4,14)(7,16)(8,15)(9,13)(10,12)(17,23,24,30)(18,27,28,32)(19,21,31,26)(20,29,22,25)(33,37,38,46)(34,40,41,48)(35,42,43,45)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8163773);