#9206394

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	#52385	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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)(3,5)(4,7)(6,12)(8,15)(9,10)(11,14)(13,16)(17,18)(19,28)(20,22)(21,26)(23,25)(24,30)(27,31)(29,32)(33,34,39,36)(35,42,41,38)(37,40,46,44)(43,48,47,45);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,14)(15,16)(17,19,23,30)(18,24,25,28)(20,29,27,26)(21,31,32,22)(33,34,39,36)(35,38,41,42)(37,44,46,40)(43,48,47,45);(2,4)(3,9)(8,12)(11,16)(18,25)(20,27)(24,28)(26,29)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,7,2)(3,10,9,5)(6,12,15,8)(11,14,16,13)(17,20,23,27)(18,22,25,31)(19,29,30,26)(21,28,32,24)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,5,7,10)(2,3,4,9)(6,14,15,13)(8,16,12,11)(17,21,23,32)(18,26,25,29)(19,22,30,31)(20,24,27,28)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,6,7,15)(2,8,4,12)(3,11,9,16)(5,13,10,14)(17,22)(18,27)(19,21)(20,25)(23,31)(24,26)(28,29)(30,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=h,e^2=gi,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, 9113590);