#408611

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#60453	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1328 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)(3,8)(4,6)(5,7)(9,10)(11,23)(12,27)(13,31)(14,21)(15,20)(16,22)(17,25)(18,29)(19,26)(24,33)(28,38)(30,37)(32,36)(34,39)(35,40);(9,11,15,13,16,26,14,25)(10,17,21,19,22,31,20,23)(12,24,29,28,30,39,32,40)(18,34,27,38,36,33,37,35);(1,3,5,4)(2,6,7,8)(9,12,15,29,16,30,14,32)(10,18,21,27,22,36,20,37)(11,24,13,28,26,39,25,40)(17,33,19,35,31,34,23,38);(1,4,5,3)(2,6,7,8)(9,13,15,26,16,25,14,11)(10,19,21,31,22,23,20,17)(12,28,29,39,30,40,32,24)(18,35,27,34,36,38,37,33)

Polycyclic presentation:

2^9|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, 315807);