#8699218

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	#52292	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4371 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8606414);