#7237082

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234437
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	#23939	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7144278);