#9290456

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

4145 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)(3,5,9,13)(4,8,11,14)(10,12,15,16)(18,19)(23,25)(24,26)(30,31)(33,34)(35,40)(36,42)(37,38)(39,41)(43,47)(44,48)(45,46);(4,11)(8,14)(10,15)(12,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,35)(34,40)(36,43)(37,39)(38,41)(42,47)(44,45)(46,48);(2,7)(3,9)(8,14)(10,15)(20,27)(23,25)(28,32)(30,31)(34,41)(35,37)(36,44)(46,47);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,36,39,44)(34,42,41,48)(35,43,37,45)(38,46,40,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37,39,35)(34,38,41,40)(36,45,44,43)(42,46,48,47);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,38)(34,35)(36,46)(37,41)(39,40)(42,43)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9197652);