#8926382

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	#47807	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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)(17,18,24,29)(19,30,32,25)(20,26,22,28)(21,27,31,23)(34,38)(39,44)(40,45)(46,48);(1,2,6,7)(3,8,9,15)(4,12,14,5)(10,11,13,16)(17,19,24,32)(18,25,29,30)(20,31,22,21)(23,26,27,28)(36,43)(40,45)(41,47)(46,48);(2,7)(5,12)(8,15)(11,16)(17,20,24,22)(18,26,29,28)(19,31,32,21)(23,25,27,30)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48);(1,3)(2,8)(4,13)(5,11)(6,9)(7,15)(10,14)(12,16)(17,21)(18,27)(19,20)(22,32)(23,29)(24,31)(25,28)(26,30)(33,35,37,42)(34,39,38,44)(36,41,43,47)(40,46,45,48);(1,4)(2,5)(3,10)(6,14)(7,12)(8,16)(9,13)(11,15)(17,22,24,20)(18,28,29,26)(19,31,32,21)(23,25,27,30)(33,36,37,43)(34,40,38,45)(35,41,42,47)(39,46,44,48);(1,5)(2,4)(3,11)(6,12)(7,14)(8,13)(9,16)(10,15)(17,23,24,27)(18,21,29,31)(19,26,32,28)(20,30,22,25)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=h,e^2=gh,f^2=g,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, 8833578);