#8944053

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3859 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851249);