#401868

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1202 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309064);