#8505334

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8412530);