#9214262

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

4147 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 9121458);