#9717454

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

3954 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)(17,18,24,28)(19,29,32,25)(20,26,21,23)(22,27,30,31)(33,34)(35,44)(36,42)(37,43)(38,41)(39,40)(45,47)(46,48);(1,3,6,10)(2,7,8,5)(4,12,14,11)(9,15,16,13)(17,19,24,32)(18,25,28,29)(20,30,21,22)(23,31,26,27)(34,40)(37,47)(41,44)(42,48);(2,8)(3,10)(9,16)(11,12)(17,20,24,21)(18,26,28,23)(19,30,32,22)(25,27,29,31)(33,35,39,38)(34,41,40,44)(36,43,46,45)(37,42,47,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,15)(14,16)(17,21,24,20)(18,23,28,26)(19,22,32,30)(25,31,29,27)(33,36,39,46)(34,42,40,48)(35,45,38,43)(37,44,47,41);(1,4,6,14)(2,9,8,16)(3,11,10,12)(5,15,7,13)(17,22)(18,27)(19,20)(21,32)(23,25)(24,30)(26,29)(28,31)(33,37)(34,43)(35,42)(36,44)(38,48)(39,47)(40,45)(41,46);(1,5,6,7)(2,10,8,3)(4,13,14,15)(9,11,16,12)(17,23)(18,20)(19,31)(21,28)(22,29)(24,26)(25,30)(27,32)(33,38,39,35)(34,44,40,41)(36,43,46,45)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=i,f^2=h,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, 9624650);