#9059230

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	#50710	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^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, 8966426);