#9063282

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	#56390	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^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, 8970478);