#9206499

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

4022 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113695);