#8159408

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	#52395	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4235 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)(17,18,23,21)(19,22,27,24)(20,28,29,25)(26,31,32,30)(33,34,39,42)(35,36,37,40)(38,46,47,41)(43,48,45,44);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19,23,27)(18,24,21,22)(20,26,29,32)(25,31,28,30)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(2,7)(3,9)(8,14)(10,15)(20,29)(25,28)(26,32)(30,31)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(1,3,6,9)(2,4,7,12)(5,10,13,15)(8,11,14,16)(17,20,23,29)(18,25,21,28)(19,26,27,32)(22,30,24,31)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,21,23,18)(19,22,27,24)(20,25,29,28)(26,31,32,30)(33,37,39,35)(34,36,42,40)(38,45,47,43)(41,44,46,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22)(18,19)(20,30)(21,27)(23,24)(25,26)(28,32)(29,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,e^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, 8066604);