#8720707

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(2,7)(3,12)(9,14)(10,13)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(2,7)(3,12)(9,14)(10,13)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,16)(10,15)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,35)(34,38)(36,41)(37,42)(39,44)(40,45)(43,47)(46,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627903);