#8873601

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,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, 8780797);