#9127680

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,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, 9034876);