#9203653

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gh,e^2=ghi,f^2=h,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, 9110849);