#8926592

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	#47809	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(34,38)(39,44)(40,45)(46,48);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,13,15,12)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(36,43)(40,45)(41,47)(46,48);(2,8)(3,9)(10,15)(11,16)(17,19,22,20)(18,23,25,26)(21,28,30,27)(24,32,29,31)(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)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(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)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(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)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(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=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, 8833788);