#8429513

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3873 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336709);