#8901415

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	#38713	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

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, 8808611);