#9718199

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

4044 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)(3,9)(4,13)(5,7)(6,10)(8,11)(12,14)(15,16)(17,18)(19,28)(20,26)(21,27)(22,25)(23,24)(29,31)(30,32)(36,44)(40,47)(42,48)(45,46);(1,3,8,5)(2,9,11,7)(4,12,6,15)(10,16,13,14)(18,24)(21,31)(25,28)(26,32)(33,34)(35,37)(36,40)(38,41)(39,43)(42,45)(44,47)(46,48);(1,4,8,6)(2,10,11,13)(3,12,5,15)(7,14,9,16)(17,19,23,22)(18,25,24,28)(20,27,30,29)(21,26,31,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,48)(45,47);(1,5,8,3)(2,7,11,9)(4,12,6,15)(10,16,13,14)(17,20,23,30)(18,26,24,32)(19,29,22,27)(21,28,31,25)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6,8,4)(2,10,11,13)(3,12,5,15)(7,16,9,14)(17,21)(18,27)(19,26)(20,28)(22,32)(23,31)(24,29)(25,30)(33,37,38,39)(34,35,41,43)(36,45,44,46)(40,42,47,48);(1,7)(2,3)(4,14)(5,11)(6,16)(8,9)(10,12)(13,15)(17,22,23,19)(18,28,24,25)(20,27,30,29)(21,26,31,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=ghi,f^2=g,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, 9625395);