#8901487

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

3912 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)(3,8)(4,6)(5,9)(7,11)(10,15)(12,13)(14,16)(33,34,40,44)(35,45,48,41)(36,39,38,42)(37,47,46,43);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30)(33,35,40,48)(34,41,44,45)(36,46,38,37)(39,47,42,43);(5,13)(9,12)(10,16)(14,15)(18,22)(23,29)(24,30)(31,32)(33,36,40,38)(34,42,44,39)(35,46,48,37)(41,43,45,47);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31)(33,37)(34,43)(35,36)(38,48)(39,41)(40,46)(42,45)(44,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32)(33,38,40,36)(34,42,44,39)(35,46,48,37)(41,47,45,43);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30)(33,39,40,42)(34,36,44,38)(35,47,48,43)(37,41,46,45)

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 8808683);