#9290406

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

4368 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,23)(19,26,25,21)(20,24,28,30)(27,32,31,29)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,45)(46,47);(2,6)(7,9)(8,13)(14,15)(18,23)(20,28)(21,26)(27,31)(33,35)(34,37)(36,44)(38,42)(39,41)(40,48)(43,46)(45,47);(4,12)(8,13)(10,16)(14,15)(18,23)(19,25)(20,28)(29,32)(34,39)(35,42)(40,47)(43,44);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19,22,25)(18,21,23,26)(20,27,28,31)(24,29,30,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,36,38,46)(34,40,39,47)(35,43,42,44)(37,45,41,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,21)(18,25)(19,23)(20,29)(22,26)(24,31)(27,30)(28,32)(33,37)(34,35)(36,45)(38,41)(39,42)(40,43)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=hi,d^2=gi,e^2=i,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, 9197602);