#6443962

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2013 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,25)(13,18)(14,19)(15,21)(17,24)(20,31)(22,32)(23,33)(26,40)(27,35)(28,36)(29,37)(30,38)(34,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,14)(13,27)(15,24)(17,21)(18,35)(19,25)(20,33)(22,28)(23,31)(26,30)(29,39)(32,36)(34,37)(38,40);(9,12,14,24,15,23,31,11)(10,17,20,25,21,16,19,33)(13,26,28,39,30,27,29,22)(18,34,37,40,38,32,36,35);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,22)(12,26)(14,28)(15,30)(16,32)(17,34)(19,36)(20,37)(21,38)(23,27)(24,39)(25,40)(29,31)(33,35);(9,14,15,31)(10,19,21,20)(11,12,24,23)(13,28,30,29)(16,25,17,33)(18,36,38,37)(22,26,39,27)(32,40,34,35)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351158);