#8838448

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

4096 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,18)(19,20)(21,29)(22,23)(24,25)(26,30)(27,31)(28,32)(33,34,38,41)(35,37,42,39)(36,40,46,48)(43,45,44,47);(1,3)(2,7)(4,5)(6,11)(8,14)(9,16)(10,12)(13,15)(18,23)(20,24)(21,30)(27,32)(35,42)(37,39)(43,44)(45,47);(3,11)(4,13)(7,12)(8,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,35)(34,39)(36,44)(37,41)(38,42)(40,45)(43,46)(47,48);(1,4,6,13)(2,8,10,16)(3,5,11,15)(7,14,12,9)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,29)(30,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,5)(2,9)(3,4)(6,15)(7,16)(8,12)(10,14)(11,13)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,35)(34,39)(36,44)(37,41)(38,42)(40,45)(43,46)(47,48);(1,2)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)(33,37,38,39)(34,35,41,42)(36,45,46,47)(40,43,48,44)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=ghi,e^2=g,f^2=h,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, 8745644);