#9713339

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=i,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, 9620535);