#8810161

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	#36890	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 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)(3,9)(4,7)(5,8)(6,10)(11,15)(12,14)(13,16)(18,21)(23,26)(24,29)(30,32)(33,34)(35,42)(36,37)(38,39)(40,45)(41,46)(43,48)(44,47);(1,2,5,8)(3,6,10,9)(4,7,12,14)(11,13,16,15)(20,28)(24,29)(25,31)(30,32)(34,39)(36,41)(40,42)(43,47);(3,10)(4,12)(6,9)(7,14)(17,18,22,21)(19,23,27,26)(20,24,28,29)(25,30,31,32)(33,34,38,39)(35,40,45,42)(36,37,41,46)(43,44,47,48);(1,3)(2,6)(4,11)(5,10)(7,13)(8,9)(12,16)(14,15)(17,19,22,27)(18,23,21,26)(20,25,28,31)(24,30,29,32)(33,35,38,45)(34,40,39,42)(36,43,41,47)(37,44,46,48);(1,4)(2,7)(3,11)(5,12)(6,13)(8,14)(9,15)(10,16)(17,20,22,28)(18,24,21,29)(19,25,27,31)(23,30,26,32)(33,36,38,41)(34,37,39,46)(35,43,45,47)(40,44,42,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21,22,18)(19,26,27,23)(20,29,28,24)(25,32,31,30)(33,37)(34,41)(35,44)(36,39)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=h,e^2=h,f^2=hi,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, 8717357);