#8458804

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 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)(3,6)(4,7)(5,8)(9,10,16,15)(11,19,14,17)(12,18,13,22)(20,24,21,23)(28,36)(32,39)(35,40)(37,38);(1,3,5,7)(2,6,8,4)(9,11,16,14)(10,17,15,19)(12,20,13,21)(18,23,22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3)(2,4)(5,7)(6,8)(9,12,16,13)(10,18,15,22)(11,20,14,21)(17,23,19,24)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,13,16,12)(10,18,15,22)(11,21,14,20)(17,23,19,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(9,14,16,11)(10,19,15,17)(12,21,13,20)(18,24,22,23)(25,28,30,36)(26,32,34,39)(27,35,33,40)(29,37,31,38);(1,4,5,6)(2,7,8,3)(9,15,16,10)(11,19,14,17)(12,22,13,18)(20,24,21,23)(25,29,30,31)(26,33,34,27)(28,37,36,38)(32,40,39,35)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=gi,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, 8366000);