#9166728

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	#55040	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4332 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,23,25)(19,27,22,24)(20,21,26,32)(28,29,30,31)(33,34,39,38)(35,43,45,40)(36,41,44,46)(37,42,47,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,22)(18,24,25,27)(20,28,26,30)(21,31,32,29)(33,35,39,45)(34,40,38,43)(36,37,44,47)(41,48,46,42);(3,10)(6,7)(11,16)(14,15)(18,25)(20,26)(24,27)(28,30)(33,36,39,44)(34,41,38,46)(35,37,45,47)(40,48,43,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,21)(19,28)(22,30)(23,26)(24,31)(25,32)(27,29)(33,36)(34,41)(35,37)(38,46)(39,44)(40,48)(42,43)(45,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21)(18,26)(19,29)(20,25)(22,31)(23,32)(24,28)(27,30)(33,37)(34,42)(35,44)(36,45)(38,48)(39,47)(40,41)(43,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,22,23,19)(18,27,25,24)(20,30,26,28)(21,29,32,31)(33,38,39,34)(35,43,45,40)(36,46,44,41)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9073924);