#9057308

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4304 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gh,e^2=g,f^2=gi,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, 8964504);