#9668267

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3805 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575463);