#9066042

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	#42052	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8973238);