#498140

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=deh,e^b=eh,h^b=hi,d^c=di,e^d=eh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405336);