#9127564

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
prime radical	#55103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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)(3,8)(4,6)(5,12)(7,11)(9,14)(10,15)(13,16)(17,18)(19,22)(20,25)(21,26)(23,24)(27,32)(28,30)(29,31)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(1,3,6,11)(2,7,4,8)(5,10,14,13)(9,15,12,16)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,30,31,26)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(2,4)(7,8)(9,12)(15,16)(18,24)(19,27)(20,29)(26,30)(36,45)(40,44)(42,48)(46,47);(1,4,6,2)(3,8,11,7)(5,12,14,9)(10,16,13,15)(17,20)(18,25)(19,28)(21,27)(22,30)(23,29)(24,31)(26,32)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,21,23,28)(18,26,24,30)(19,29,27,20)(22,31,32,25)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,22)(18,27)(19,24)(20,30)(21,25)(23,32)(26,29)(28,31)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9034760);