#8720301

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=ghi,e^2=g,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, 8627497);