#9713396

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,20)(19,27,28,24)(21,25,32,29)(22,30,31,26)(33,34)(35,42)(36,39)(37,40)(38,44)(41,46)(43,47)(45,48);(4,11)(8,14)(10,15)(12,16)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,30,29,26)(35,37)(40,42)(43,45)(47,48);(2,7)(3,9)(8,14)(10,15)(18,20)(24,27)(25,29)(26,30)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,18)(19,27,28,24)(21,29,32,25)(22,30,31,26)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21,23,32)(18,25,20,29)(19,22,28,31)(24,26,27,30)(33,37,39,35)(34,40,36,42)(38,43,46,45)(41,48,44,47);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,26)(19,21)(20,30)(23,31)(24,25)(27,29)(28,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9620592);