#8437805

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	#51061	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4334 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8345001);