#9713374

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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)(19,28)(20,26)(21,27)(22,24)(23,25)(29,32)(30,31)(36,44)(40,47)(43,45)(46,48);(4,11)(8,14)(10,15)(12,16)(17,19,23,22)(18,24,25,28)(20,29,27,31)(21,32,26,30)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(2,7)(3,9)(8,14)(10,15)(18,25)(20,27)(24,28)(29,31)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,46,47,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,27)(18,26,25,21)(19,29,22,31)(24,30,28,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,26)(18,27,25,20)(19,30,22,32)(24,29,28,31)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,19)(18,28,25,24)(20,31,27,29)(21,30,26,32)(33,37,38,35)(34,39,41,42)(36,45,44,43)(40,46,47,48)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=g,f^2=gi,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, 9620570);