#9545802

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

4141 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=i,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9452998);