#9240943

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
prime radical	#55299	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=h,e^2=gh,f^2=h,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, 9148139);