#8709127

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

4166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8616323);