#8159585

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

4220 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066781);