#400651

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

802 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi,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, 307847);