#9189554

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	#46575	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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)(4,13)(5,9)(7,8)(10,16)(11,12)(14,15)(17,18,23,27)(19,24,30,32)(20,25,31,22)(21,26,29,28)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,44)(47,48);(1,3,7,10)(2,6,8,16)(4,11,9,15)(5,14,13,12)(17,19,23,30)(18,24,27,32)(20,28,31,26)(21,22,29,25)(33,34,38,41)(35,36,39,42)(37,40,46,45)(43,44,47,48);(2,8)(3,10)(4,9)(12,14)(17,20,23,31)(18,25,27,22)(19,26,30,28)(21,24,29,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,4)(2,5)(3,11)(6,14)(7,9)(8,13)(10,15)(12,16)(17,20)(18,22)(19,28)(21,32)(23,31)(24,29)(25,27)(26,30)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,5)(2,9)(3,12)(4,8)(6,11)(7,13)(10,14)(15,16)(17,21)(18,26)(19,22)(20,24)(23,29)(25,30)(27,28)(31,32)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47);(1,6)(2,10)(3,8)(4,14)(5,15)(7,16)(9,12)(11,13)(17,22)(18,20)(19,29)(21,30)(23,25)(24,28)(26,32)(27,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^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, 9096750);