#8901331

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

3925 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)(19,23)(20,29)(21,24)(22,26)(25,28)(27,32)(30,31)(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)(18,23)(20,30)(21,27)(22,28)(24,32)(25,26)(29,31)(33,35,40,48)(34,41,44,45)(36,46,38,37)(39,47,42,43);(2,6)(7,13)(8,14)(15,16)(17,20,22,21)(18,24,26,29)(19,27,28,30)(23,31,25,32)(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)(17,21,22,20)(18,24,26,29)(19,27,28,30)(23,32,25,31)(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)(17,18)(19,25)(20,24)(21,29)(22,26)(23,28)(27,32)(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)(17,19)(18,25)(20,30)(21,27)(22,28)(23,26)(24,31)(29,32)(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=h,d^2=i,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, 8808527);