#9173185

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

4128 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,14)(12,16)(17,18,22,26)(19,23,27,24)(20,21,25,31)(28,29,30,32)(33,34)(35,37)(36,40)(38,41)(39,43)(42,46)(44,47)(45,48);(2,7)(4,12)(8,14)(10,13)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,30)(31,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,45)(44,48)(46,47);(2,7)(3,11)(9,16)(10,13)(19,27)(23,24)(28,30)(29,32)(36,44)(40,47)(42,48)(45,46);(1,3)(2,8)(4,13)(5,9)(6,11)(7,14)(10,12)(15,16)(17,19)(18,24)(20,30)(21,29)(22,27)(23,26)(25,28)(31,32)(33,36)(34,40)(35,42)(37,46)(38,44)(39,45)(41,47)(43,48);(1,4)(2,9)(3,10)(5,14)(6,12)(7,16)(8,15)(11,13)(17,20)(18,21)(19,28)(22,25)(23,29)(24,32)(26,31)(27,30)(33,37,38,39)(34,35,41,43)(36,45,44,46)(40,48,47,42);(1,5,6,15)(2,10,7,13)(3,9,11,16)(4,8,12,14)(17,21)(18,25)(19,29)(20,26)(22,31)(23,30)(24,28)(27,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,46)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,e^2=gh,f^2=h,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, 9080381);