#502155

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

817 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,e^d=eh,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, 409351);