#8901584

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	#46494	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,34,39,44)(35,43,46,40)(36,47,48,41)(37,38,45,42);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35,39,46)(34,40,44,43)(36,45,48,37)(38,41,42,47);(2,6)(7,13)(8,14)(15,16)(17,19,22,20)(18,24,26,23)(21,27,29,28)(25,31,32,30)(33,36,39,48)(34,41,44,47)(35,45,46,37)(38,40,42,43);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,37)(34,42)(35,36)(38,44)(39,45)(40,47)(41,43)(46,48);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,35)(34,43)(36,45)(37,48)(38,41)(39,46)(40,44)(42,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,38,39,42)(34,37,44,45)(35,41,46,47)(36,43,48,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=ghi,e^2=i,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, 8808780);