#8505034

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	#37331	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4401 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=hi,e^2=g,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, 8412230);