#8662912

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	#49219	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570108);