#7065221

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

1726 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=fi,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, 6972417);