#9711928

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

3949 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,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, 9619124);