#8197249

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	#57732	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104445);