#7007211

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,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, 6914407);