#8772531

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

3850 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)(3,11)(4,14)(5,15)(6,10)(7,13)(8,12)(9,16)(17,18,22,23)(19,25,28,24)(20,21,29,30)(26,31,32,27)(35,37)(39,41)(43,45)(47,48);(1,3)(2,7)(4,5)(6,13)(8,15)(9,14)(10,11)(12,16)(18,23)(20,29)(24,25)(26,32)(33,34,38,42)(35,39,37,41)(36,44,46,40)(43,48,45,47);(1,2,6,10)(3,7,13,11)(4,14,12,8)(5,9,16,15)(17,18)(19,24)(20,21)(22,23)(25,28)(26,27)(29,30)(31,32)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(1,4,6,12)(2,8,10,14)(3,5,13,16)(7,15,11,9)(17,19,22,28)(18,24,23,25)(20,26,29,32)(21,27,30,31);(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,14)(15,16)(17,20)(18,21)(19,26)(22,29)(23,30)(24,27)(25,31)(28,32)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,5)(2,9)(3,12)(4,13)(6,16)(7,8)(10,15)(11,14)(17,21)(18,20)(19,27)(22,30)(23,29)(24,26)(25,32)(28,31)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679727);