#8901563

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	#46538	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=i,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, 8808759);