#8436539

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	#41410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4498 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=gh,f^2=g,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, 8343735);