#9183910

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	#50739	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=gh,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, 9091106);