#8901655

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

3914 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)(19,24)(20,26)(21,27)(22,30)(23,25)(28,31)(29,32)(33,34,40,44)(35,45,48,41)(36,39,38,42)(37,47,46,43);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19)(18,24)(20,29)(21,22)(23,28)(25,31)(26,32)(27,30)(33,35,40,48)(34,41,44,45)(36,46,38,37)(39,47,42,43);(2,6)(7,13)(8,14)(15,16)(18,25)(19,28)(22,26)(27,29)(33,36,40,38)(34,42,44,39)(35,46,48,37)(41,43,45,47);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,20)(18,26)(19,29)(21,31)(22,25)(23,30)(24,32)(27,28)(33,37)(34,43)(35,36)(38,48)(39,41)(40,46)(42,45)(44,47);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,21)(18,27)(19,26)(20,24)(22,28)(23,32)(25,29)(30,31)(33,38,40,36)(34,42,44,39)(35,46,48,37)(41,47,45,43);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,22,23,26)(18,20,25,30)(19,21,28,32)(24,29,31,27)(33,39,40,42)(34,36,44,38)(35,47,48,43)(37,41,46,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=ghi,e^2=hi,f^2=h,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, 8808851);