#8159659

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	#57484	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=i,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,d^c=dgi,e^d=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, 8066855);