#7381631

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1487 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,32,37)(27,33,38,34)(28,39,29,35)(30,36,31,40);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,32,38)(26,33,37,34)(28,30,29,31)(35,40,39,36);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,34)(27,31)(29,32)(30,38)(33,37)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(11,18)(12,20)(14,22)(15,24)(25,29,32,28)(26,35,37,39)(27,30,38,31)(33,36,34,40);(2,5)(6,8)(11,18)(14,22)(17,19)(21,23)(25,30,32,31)(26,36,37,40)(27,28,38,29)(33,39,34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=fi,e^2=i,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,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7288827);