#9016622

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	#53532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8923818);