#498139

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

764 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)(10,16)(11,15)(13,26)(17,32)(18,22)(19,31)(20,23)(21,29)(24,28)(27,30)(33,38)(34,35)(36,39)(37,40);(5,7,8,6)(9,10)(11,17)(12,22)(13,27)(14,19)(15,30)(16,26)(18,28)(20,35)(21,31)(23,38)(24,34)(25,39)(29,32)(33,37)(36,40);(2,4)(9,11,14,15)(10,17,19,30)(12,23,25,20)(13,21,29,26)(16,27,31,32)(18,34,36,33)(22,38,39,35)(24,40,37,28);(9,12,14,25)(10,18,19,36)(11,20,15,23)(13,28,29,40)(16,22,31,39)(17,33,30,34)(21,37,26,24)(27,35,32,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,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, 405335);