#8726488

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

3937 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633684);