#8197259

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	#56192	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(17,18,22,23)(19,25,27,21)(20,28,30,24)(26,29,32,31);(2,7)(3,9)(10,14)(11,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,31)(30,32)(33,34,38,37)(35,41,43,39)(36,40,45,44)(42,47,48,46);(5,13)(10,14)(11,16)(12,15)(18,23)(21,25)(24,28)(29,31)(34,37)(39,41)(40,44)(46,47);(1,3,6,9)(2,8,7,4)(5,11,13,16)(10,15,14,12)(17,20,22,30)(18,24,23,28)(19,26,27,32)(21,29,25,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,31)(30,32)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,21)(18,19)(20,29)(22,25)(23,27)(24,26)(28,32)(30,31)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46)

Polycyclic presentation:

2^9|a^2=g,b^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, 8104455);