#9236462

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143658);