#9923661

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

4098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830857);