#9030144

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	#53694	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(18,23)(19,25)(26,30)(27,32)(33,34,39,42)(35,40,38,36)(37,41,47,48)(43,45,46,44);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,39,38)(34,40,42,36)(37,43,47,46)(41,45,48,44);(2,6)(3,5)(4,12)(15,16)(19,25)(20,24)(21,29)(26,30)(33,36,39,40)(34,35,42,38)(37,45,47,44)(41,46,48,43);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,36)(34,35)(37,45)(38,42)(39,40)(41,46)(43,48)(44,47);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,38,39,35)(34,36,42,40)(37,46,47,43)(41,44,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8937340);