#498137

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

788 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)(9,10)(11,17)(12,15)(13,16)(14,27)(18,24)(19,25)(20,30)(21,32)(22,34)(23,35)(26,29)(28,33)(31,38)(36,39)(37,40);(5,7,8,6)(10,15)(11,20)(12,24)(14,25)(16,29)(18,19)(21,36)(22,23)(26,38)(27,31)(28,37)(30,34)(33,35)(39,40);(2,4)(10,16)(11,21)(15,29)(18,31)(19,27)(20,36)(22,37)(23,28);(9,11,13,21)(10,17,16,32)(12,20,26,36)(14,28,25,23)(15,30,29,39)(18,34,31,40)(19,35,27,33)(22,38,37,24)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,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, 405333);