#9726627

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

4087 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)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18)(19,20)(21,29)(22,24)(23,27)(25,31)(26,32)(28,30)(34,39)(35,42)(37,45)(44,47);(1,3,7,4)(2,8,9,13)(5,14,15,11)(6,12,16,10)(17,19,22,27)(18,23,24,20)(21,26,31,30)(25,32,29,28)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,9)(3,4)(10,12)(11,14)(18,24)(20,23)(25,29)(28,32)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,4,7,3)(2,8,9,13)(5,14,15,11)(6,10,16,12)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,30)(31,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,5,7,15)(2,10,9,12)(3,11,4,14)(6,8,16,13)(17,19)(18,20)(21,30)(22,27)(23,24)(25,32)(26,31)(28,29)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,6,7,16)(2,11,9,14)(3,12,4,10)(5,13,15,8)(17,21)(18,25)(19,26)(20,28)(22,31)(23,32)(24,29)(27,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9633823);