#8806712

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

4194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8713908);