#8197341

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	#55126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=g,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, 8104537);