#7232017

order	512
nilpotency class	3
isoclinism	isoclinic class of #7199982
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 ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1596 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)(11,17)(12,20)(13,15)(14,18)(16,22)(19,23)(21,24)(25,26)(27,34)(28,32)(29,31)(30,35)(33,37)(36,39)(38,40);(2,5)(3,7)(9,11)(10,14)(12,21)(13,18)(15,17)(16,24)(19,22)(20,23)(25,27,28,35)(26,30,32,34)(29,36,37,40)(31,38,33,39);(11,18)(14,17)(19,21)(23,24)(26,31)(29,37)(30,38)(32,33)(34,39)(36,40);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,20)(23,24)(27,35)(30,34)(36,40)(38,39);(2,5)(6,8)(9,12,13,22)(10,16,15,20)(11,19,18,21)(14,23,17,24)(26,32)(30,34)(31,33)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7139213);