#9206648

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=h,e^2=ghi,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, 9113844);