#9713367

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3894 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)(3,13)(4,10)(5,11)(6,16)(7,9)(8,14)(12,15)(19,22)(20,24)(27,31)(28,32)(36,44)(39,46)(43,47)(45,48);(1,3,7,14)(2,8,9,13)(4,12,11,16)(5,15,10,6)(17,18)(19,20)(21,29)(22,24)(23,26)(25,30)(27,32)(28,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(2,9)(4,11)(8,13)(12,16)(17,19,23,22)(18,24,26,20)(21,27,30,31)(25,32,29,28)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,45,46,48);(1,4,7,11)(2,10,9,5)(3,12,14,16)(6,13,15,8)(17,20,23,24)(18,19,26,22)(21,28,30,32)(25,27,29,31)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,46)(42,48);(1,5,7,10)(2,11,9,4)(3,6,14,15)(8,12,13,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,30)(24,32)(26,29)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48);(1,6)(2,12)(3,5)(4,13)(7,15)(8,11)(9,16)(10,14)(17,22,23,19)(18,24,26,20)(21,27,30,31)(25,28,29,32)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,46)(44,48)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=g,f^2=gh,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, 9620563);