#409832

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

806 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:

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

Polycyclic presentation:

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

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, 317028);