#8688398

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	#44873	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595594);