#9240975

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

3997 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=h,e^2=gh,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, 9148171);