#9712492

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	#51040	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9619688);