#8908348

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

4158 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,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, 8815544);