#9240972

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

3978 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=gh,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, 9148168);