#9726445

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3809 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18,22,21)(19,25,27,23)(20,24,29,28)(26,31,32,30)(33,34,39,45)(35,44,38,40)(36,42,47,41)(37,43,48,46);(1,3,7,4)(2,8,9,13)(5,14,15,11)(6,12,16,10)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35,39,38)(34,40,45,44)(36,43,47,46)(37,41,48,42);(2,9)(3,4)(10,12)(11,14)(33,36,39,47)(34,41,45,42)(35,43,38,46)(37,40,48,44);(1,4,7,3)(2,8,9,13)(5,14,15,11)(6,10,16,12)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,36)(34,42)(35,43)(37,40)(38,46)(39,47)(41,45)(44,48);(1,5,7,15)(2,10,9,12)(3,11,4,14)(6,8,16,13)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,37)(34,43)(35,42)(36,40)(38,41)(39,48)(44,47)(45,46);(1,6,7,16)(2,11,9,14)(3,12,4,10)(5,13,15,8)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,38,39,35)(34,44,45,40)(36,46,47,43)(37,42,48,41)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,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, 9633641);