#9054388

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

3831 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961584);