#9712814

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3874 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)(3,10)(4,7)(5,8)(6,12)(9,14)(11,15)(13,16)(17,18,24,21)(19,27,23,25)(20,29,22,26)(28,31,30,32)(34,37)(39,41)(40,45)(46,47);(3,5)(8,10)(11,13)(15,16)(17,19,24,23)(18,25,21,27)(20,28,22,30)(26,31,29,32)(36,44)(40,45)(42,48)(46,47);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,20,24,22)(18,26,21,29)(19,28,23,30)(25,31,27,32)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,21,24,18)(19,27,23,25)(20,26,22,29)(28,31,30,32)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,5,7,3)(2,8,4,10)(6,11,14,13)(9,16,12,15)(17,22,24,20)(18,26,21,29)(19,28,23,30)(25,32,27,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,23,24,19)(18,27,21,25)(20,30,22,28)(26,32,29,31)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gi,f^2=hi,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, 9620010);