#8722263

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	#42629	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4073 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)(3,11)(4,13)(5,8)(6,12)(7,9)(10,15)(14,16)(17,18,23,22)(19,24,27,28)(20,25,29,31)(21,26,30,32)(33,34)(35,44)(36,37)(38,42)(39,43)(40,47)(41,46)(45,48);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,10,16,13)(17,19)(18,24)(20,21)(22,28)(23,27)(25,26)(29,30)(31,32)(33,35)(34,40)(36,45)(37,42)(38,46)(39,47)(41,48)(43,44);(2,9)(4,15)(8,11)(12,14)(19,27)(20,29)(24,28)(25,31)(33,36,39,46)(34,37,43,41)(35,45,47,38)(40,42,44,48);(1,4)(2,10)(3,12)(5,14)(6,8)(7,15)(9,13)(11,16)(17,20)(18,25)(19,21)(22,31)(23,29)(24,26)(27,30)(28,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,10,16,13)(17,21)(18,26)(19,20)(22,32)(23,30)(24,25)(27,29)(28,31)(33,37,39,41)(34,36,43,46)(35,42,47,48)(38,40,45,44);(1,6,7,16)(2,12,9,14)(3,10,5,13)(4,11,15,8)(17,22,23,18)(19,28,27,24)(20,25,29,31)(21,26,30,32)(33,38,39,45)(34,42,43,48)(35,46,47,36)(37,44,41,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8629459);