#390640

order	512
nilpotency class	3
isoclinism	isoclinic class of #219706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

775 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=gh,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=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, 297836);