#8926585

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	#57384	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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)(3,7,9,4)(5,10,14,15)(11,13,16,12)(17,18,23,28)(19,27,22,24)(20,21,26,25)(29,32,31,30)(34,38)(39,44)(40,45)(46,48);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,13,15,12)(17,19,23,22)(18,24,28,27)(20,31,26,29)(21,32,25,30)(36,43)(40,45)(41,47)(46,48);(2,8)(3,9)(10,15)(11,16)(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,32,27,30)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48);(1,4,6,7)(2,9,8,3)(5,13,14,12)(10,11,15,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,32)(30,31)(33,35,37,42)(34,39,38,44)(36,41,43,47)(40,46,45,48);(1,5,6,14)(2,10,8,15)(3,11,9,16)(4,12,7,13)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,31,27,29)(33,36,37,43)(34,40,38,45)(35,41,42,47)(39,46,44,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,16)(14,15)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,32,25,30)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gh,e^2=h,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, 8833781);