#9717567

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	#38816	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,28)(20,30)(21,25)(22,29)(23,24)(26,31)(27,32)(36,44)(39,46)(42,48)(45,47);(1,3)(2,6)(4,10)(5,13)(7,9)(8,14)(11,15)(12,16)(18,24)(19,21)(26,30)(27,29)(33,34)(35,40)(36,39)(37,41)(38,43)(42,47)(44,46)(45,48);(2,4)(3,9)(8,12)(11,13)(17,19,23,21)(18,25,24,28)(20,27,31,29)(22,26,32,30)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4,7,2)(3,10,9,6)(5,12,15,8)(11,14,13,16)(17,20,23,31)(18,26,24,30)(19,29,21,27)(22,28,32,25)(33,36)(34,39)(35,42)(37,44)(38,45)(40,47)(41,46)(43,48);(1,5,7,15)(2,8,4,12)(3,11,9,13)(6,16,10,14)(17,21,23,19)(18,25,24,28)(20,27,31,29)(22,30,32,26)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,6,7,10)(2,9,4,3)(5,14,15,16)(8,11,12,13)(17,22)(18,27)(19,26)(20,28)(21,30)(23,32)(24,29)(25,31)(33,34)(35,38)(36,39)(37,41)(40,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=h,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, 9624763);