#403152

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1200 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)(9,10,14,17,13,18,24,28)(11,19,21,33,22,15,27,30)(12,23,26,37,25,16,31,32)(20,29,35,39,36,34,38,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,27,13,22,24,21)(10,15,28,30,18,19,17,33)(12,20,26,38,25,36,31,35)(16,29,37,39,23,34,32,40);(3,7)(5,8)(9,10)(11,15)(12,16)(13,18)(14,28)(17,24)(19,22)(20,29)(21,33)(23,25)(26,37)(27,30)(31,32)(34,36)(35,40)(38,39);(9,12,24,31,13,25,14,26)(10,16,17,32,18,23,28,37)(11,20,21,35,22,36,27,38)(15,29,33,40,19,34,30,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=efi,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=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, 310348);