#9127671

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	#44129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4132 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)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(1,3)(2,8)(4,6)(5,13)(7,12)(9,10)(11,15)(14,16)(17,19,22,24)(18,21,25,26)(20,27,29,31)(23,30,28,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(3,12)(4,14)(8,15)(9,13)(20,29)(23,28)(27,31)(30,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(1,4)(2,9)(3,6)(5,15)(7,14)(8,10)(11,13)(12,16)(17,20,22,29)(18,23,25,28)(19,27,24,31)(21,30,26,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(1,5)(2,6)(3,13)(4,8)(7,10)(9,12)(11,16)(14,15)(17,18)(19,21)(20,28)(22,25)(23,29)(24,26)(27,32)(30,31)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,6)(2,10)(3,4)(5,11)(7,16)(8,9)(12,14)(13,15)(17,21)(18,24)(19,25)(20,30)(22,26)(23,31)(27,28)(29,32)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=g,e^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, 9034867);