#483112

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60649	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1031 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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=hi,f^2=gh,h^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390308);