#9164015

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	#41815	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4124 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)(3,12)(4,14)(5,11)(6,8)(7,9)(10,15)(13,16)(17,18,23,21)(19,24,27,29)(20,25,31,30)(22,26,32,28)(33,34)(35,42)(36,41)(37,40)(38,43)(39,44)(45,48)(46,47);(1,3,7,6)(2,8,9,12)(4,13,15,11)(5,10,16,14)(17,19,23,27)(18,24,21,29)(20,28,31,26)(22,30,32,25)(33,35,39,37)(34,40,44,42)(36,38,45,46)(41,47,48,43);(2,9)(4,15)(8,12)(11,13)(19,27)(20,31)(24,29)(25,30)(33,36,39,45)(34,41,44,48)(35,38,37,46)(40,47,42,43);(1,4)(2,10)(3,13)(5,12)(6,11)(7,15)(8,16)(9,14)(17,20)(18,25)(19,28)(21,30)(22,24)(23,31)(26,27)(29,32)(33,37,39,35)(34,42,44,40)(36,38,45,46)(41,47,48,43);(1,5,7,16)(2,11,9,13)(3,14,6,10)(4,8,15,12)(17,21,23,18)(19,29,27,24)(20,25,31,30)(22,26,32,28)(33,38)(34,43)(35,45)(36,37)(39,46)(40,41)(42,48)(44,47);(1,6,7,3)(2,12,9,8)(4,11,15,13)(5,14,16,10)(17,22)(18,26)(19,30)(20,24)(21,28)(23,32)(25,27)(29,31)(33,34)(35,40)(36,41)(37,42)(38,47)(39,44)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=gh,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, 9071211);