#9183848

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

4127 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,13)(14,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,47)(45,46);(1,3,6,5)(2,7,9,10)(4,11,12,14)(8,15,13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35)(34,38)(36,44)(37,43)(39,48)(40,41)(42,46)(45,47);(4,12)(8,13)(11,14)(15,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,34,37,40)(35,38,43,41)(36,39,46,45)(42,47,44,48);(1,4,6,12)(2,8,9,13)(3,11,5,14)(7,15,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,34)(35,38)(36,45)(37,40)(39,46)(41,43)(42,48)(44,47);(1,2)(3,7)(4,13)(5,10)(6,9)(8,12)(11,16)(14,15)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,36,37,46)(34,39,40,45)(35,42,43,44)(38,47,41,48);(1,5,6,3)(2,7,9,10)(4,14,12,11)(8,15,13,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,35)(34,38)(36,44)(37,43)(39,48)(40,41)(42,46)(45,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gh,e^2=h,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, 9091044);