#9173159

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

4279 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:

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

Polycyclic presentation:

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

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, 9080355);