#6564832

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6472028);