#9712858

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

4044 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=g,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, 9620054);