#8075564

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4253 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982760);