#8767225

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	#45974	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8674421);