#8119260

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	#56389	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8026456);