#8926450

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	#58507	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833646);