#7063123

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
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
prime radical	#61194	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 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)(9,10,14,19,15,11,20,18)(12,23,26,21,27,34,25,16)(13,17,30,28,31,22,29,33)(24,36,38,35,39,37,40,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,21)(13,28)(14,18)(16,27)(17,29)(19,20)(22,30)(23,25)(24,37)(26,34)(31,33)(32,38)(35,40)(36,39);(1,3,4,8)(2,5,6,7)(9,12,15,27)(10,16,11,21)(13,24,31,39)(14,25,20,26)(17,32,22,35)(18,23,19,34)(28,37,33,36)(29,38,30,40);(3,8)(5,7)(9,13)(10,17)(11,22)(12,24)(14,30)(15,31)(16,32)(18,33)(19,28)(20,29)(21,35)(23,36)(25,40)(26,38)(27,39)(34,37);(2,6)(5,7)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6970319);