#9712477

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=h,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, 9619673);