#7339913

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7247109);