#7333246

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1294 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7240442);