#9214256

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

4001 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9121452);