#8837061

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

4117 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)(3,5)(4,9)(6,14)(7,8)(10,15)(11,16)(12,13)(17,18,22,24)(19,26,29,23)(20,30,25,21)(27,32,31,28)(33,34)(35,37)(36,44)(38,39)(40,43)(41,47)(42,48)(45,46);(1,3,7,4)(2,5,8,9)(6,12,15,11)(10,16,14,13)(17,19,22,29)(18,23,24,26)(20,27,25,31)(21,32,30,28)(34,39)(37,40)(41,44)(46,48);(2,8)(3,4)(10,14)(11,12)(18,24)(20,25)(23,26)(27,31)(33,35,38,43)(34,37,39,40)(36,42,47,45)(41,46,44,48);(1,4,7,3)(2,9,8,5)(6,12,15,11)(10,16,14,13)(17,20,22,25)(18,21,24,30)(19,27,29,31)(23,32,26,28)(33,35)(34,40)(36,45)(37,39)(38,43)(41,46)(42,47)(44,48);(1,5,7,9)(2,4,8,3)(6,13,15,16)(10,12,14,11)(17,19)(18,23)(20,31)(21,28)(22,29)(24,26)(25,27)(30,32)(33,36,38,47)(34,41,39,44)(35,42,43,45)(37,46,40,48);(1,6,7,15)(2,10,8,14)(3,11,4,12)(5,13,9,16)(17,21)(18,25)(19,28)(20,24)(22,30)(23,27)(26,31)(29,32)(33,37,38,40)(34,35,39,43)(36,46,47,48)(41,42,44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8744257);