#498147

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	C₂×#2186	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

808 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)(13,25)(17,33)(18,31)(19,32)(20,30)(21,28)(22,37)(23,27)(26,29)(34,39)(35,40)(36,38);(5,7,8,6)(9,10)(11,17)(12,22)(13,26)(14,19)(15,29)(16,25)(18,35)(20,39)(21,32)(23,34)(24,31)(27,37)(28,33)(30,36)(38,40);(2,4)(9,11,14,15)(10,17,19,29)(12,20,24,30)(13,21,28,25)(16,26,32,33)(18,34,37,38)(22,39,31,36)(23,27,40,35);(9,12,14,24)(10,18,19,37)(11,20,15,30)(13,27,28,35)(16,31,32,22)(17,34,29,38)(21,40,25,23)(26,36,33,39)

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=dehi,e^b=eh,h^b=hi,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, 405343);