#8718316

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	#57208	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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)(17,18,23,20)(19,24,25,21)(22,26,31,28)(27,30,32,29)(33,34,39,43)(35,40,44,47)(36,41,45,46)(37,38,42,48);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,10,16,13)(17,19,23,25)(18,24,20,21)(22,27,31,32)(26,30,28,29)(33,35,39,44)(34,40,43,47)(36,37,45,42)(38,46,48,41);(2,9)(4,15)(8,11)(12,14)(18,20)(19,25)(26,28)(27,32)(35,44)(36,45)(40,47)(41,46);(1,4)(2,10)(3,12)(5,14)(6,8)(7,15)(9,13)(11,16)(17,20,23,18)(19,24,25,21)(22,26,31,28)(27,29,32,30)(33,36,39,45)(34,41,43,46)(35,37,44,42)(38,47,48,40);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,10,16,13)(17,21,23,24)(18,25,20,19)(22,30,31,29)(26,27,28,32)(33,37)(34,38)(35,45)(36,44)(39,42)(40,46)(41,47)(43,48);(1,6,7,16)(2,12,9,14)(3,10,5,13)(4,11,15,8)(17,22,23,31)(18,26,20,28)(19,27,25,32)(21,29,24,30)(33,38)(34,42)(35,46)(36,40)(37,43)(39,48)(41,44)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=i,e^2=h,f^2=h,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, 8625512);