#9063293

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	#35437	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4082 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8970489);