#9712867

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3810 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=i,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, 9620063);