#9668264

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	#2456	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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,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, 9575460);