#8676891

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	#42617	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4442 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8584087);