#8196949

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34,40,37)(35,41,38,47)(36,45,44,42)(39,43,46,48);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,40,38)(34,41,37,47)(36,39,44,46)(42,48,45,43);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,36)(34,42)(35,39)(37,45)(38,46)(40,44)(41,48)(43,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,37,40,34)(35,41,38,47)(36,45,44,42)(39,48,46,43);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,38,40,35)(34,41,37,47)(36,46,44,39)(42,48,45,43);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,39,40,46)(34,43,37,48)(35,44,38,36)(41,42,47,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=ghi,e^2=h,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, 8104145);