#8720429

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gi,e^2=h,f^2=h,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, 8627625);