#8227349

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	#54285	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32);(2,6)(7,9)(8,13)(14,15)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(4,12)(8,13)(10,16)(14,15)(18,21)(23,25)(24,28)(31,32)(34,39)(37,46)(40,43)(44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,32)(29,30)(33,36,38,45)(34,40,39,43)(35,42,41,47)(37,44,46,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8134545);