#498564

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×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

716 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fi,d^2=ehi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=eh,g^b=gh,h^b=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, 405760);