#9164009

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	#48407	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18,23,21)(19,24,27,29)(20,25,31,30)(22,26,32,28)(33,34)(35,41)(36,39)(37,42)(38,44)(40,45)(43,47)(46,48);(2,7)(4,8)(9,13)(12,14)(17,19,23,27)(18,24,21,29)(20,28,31,26)(22,30,32,25)(33,35)(34,38)(36,43)(37,44)(39,46)(40,47)(41,42)(45,48);(2,7)(3,10)(9,13)(11,16)(19,27)(20,31)(24,29)(25,30)(33,36,37,45)(34,39,42,40)(35,43,44,48)(38,46,41,47);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,20)(18,25)(19,28)(21,30)(22,24)(23,31)(26,27)(29,32)(33,36)(34,40)(35,43)(37,45)(38,47)(39,42)(41,46)(44,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,21,23,18)(19,29,27,24)(20,25,31,30)(22,26,32,28)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,46)(47,48);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,22)(18,26)(19,30)(20,24)(21,28)(23,32)(25,27)(29,31)(33,35)(34,41)(36,43)(37,44)(38,42)(39,47)(40,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=h,e^2=hi,f^2=h,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, 9071205);