#8158721

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	#52387	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=g,f^2=h,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, 8065917);