#481475

order	512
nilpotency class	4
isoclinism	isoclinic class of #481466
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60458	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di

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, 388671);