#9167606

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

4054 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(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)(18,20)(23,24)(25,28)(30,31)(33,35,39,45)(34,40,38,43)(36,37,44,47)(41,48,46,42);(3,10)(6,7)(11,16)(14,15)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(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)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(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)(17,20,22,18)(19,24,27,23)(21,25,29,28)(26,30,32,31)(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)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(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=gh,c^2=g,e^2=h,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, 9074802);