#8159244

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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)(3,5)(4,12)(6,7)(8,14)(9,10)(11,16)(13,15)(19,20)(23,24)(27,28)(30,31)(33,34,38,42)(35,39,36,40)(37,45,46,41)(43,48,44,47);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(2,7)(3,9)(8,12)(11,15)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,2)(3,10)(4,12)(5,9)(6,7)(8,14)(11,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,48)(44,47)(45,46);(1,4,6,14)(2,8,7,12)(3,11,9,15)(5,13,10,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,5,6,10)(2,9,7,3)(4,13,14,16)(8,15,12,11)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,e^2=gh,f^2=gh,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, 8066440);