#8873647

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	#55457	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8780843);