#9379992

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

4175 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9287188);