#8901571

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

3909 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gi,e^2=gi,f^2=gh,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, 8808767);