#8934885

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

3967 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18)(19,23)(20,29)(21,26)(22,27)(24,28)(25,31)(30,32)(33,34,39,41)(35,40,44,48)(36,37,47,42)(38,46,45,43);(2,6)(7,9)(8,12)(14,15)(17,19,22,28)(18,23,27,24)(20,30,31,26)(21,29,32,25)(33,35)(34,40)(36,46)(37,45)(38,42)(39,44)(41,48)(43,47);(2,6)(4,13)(7,9)(10,16)(19,28)(21,32)(23,24)(26,30)(34,41)(35,44)(36,47)(38,45);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,24)(20,30)(21,29)(22,28)(23,27)(25,32)(26,31)(33,36,39,47)(34,42,41,37)(35,43,44,46)(38,40,45,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,22,31)(18,25,27,29)(19,26,28,30)(21,23,32,24)(33,37,39,42)(34,36,41,47)(35,45,44,38)(40,46,48,43);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21)(18,26)(19,29)(20,23)(22,32)(24,31)(25,28)(27,30)(33,38,39,45)(34,43,41,46)(35,42,44,37)(36,40,47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842081);