#9127597

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3973 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,e^2=gh,f^2=g,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, 9034793);