#8818008

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	#59854	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(3,6)(4,11)(5,7)(8,13)(9,15)(10,16)(12,14)(33,34,38,39)(35,42,40,36)(37,41,45,46)(43,48,47,44);(1,3,5,9)(2,6,7,15)(4,12,13,10)(8,16,11,14)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(34,39)(35,40)(37,45)(44,48);(2,7)(3,9)(8,11)(10,12)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,35,38,40)(34,36,39,42)(37,43,45,47)(41,44,46,48);(1,2)(3,6)(4,8)(5,7)(9,15)(10,14)(11,13)(12,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48);(1,3)(2,6)(4,10)(5,9)(7,15)(8,14)(11,16)(12,13)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,4,5,13)(2,8,7,11)(3,10,9,12)(6,14,15,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725204);