#8159092

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,e^2=h,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, 8066288);