#409342

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 316538);