#8901474

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3923 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=ghi,e^2=hi,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, 8808670);