#408610

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
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

1342 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)(11,19)(12,26)(13,17)(14,22)(15,20)(16,21)(18,29)(23,34)(24,32)(25,33)(27,35)(28,37)(30,36)(31,39)(38,40);(9,11,15,24,16,25,14,13)(10,17,22,33,21,32,20,19)(12,23,28,38,30,39,29,27)(18,34,36,35,37,31,26,40);(1,3,5,4)(2,6,7,8)(9,12,14,29,16,30,15,28)(10,18,20,26,21,37,22,36)(11,23,13,27,25,39,24,38)(17,31,19,35,32,34,33,40);(1,4,5,3)(2,6,7,8)(9,13,14,25,16,24,15,11)(10,19,20,32,21,33,22,17)(12,27,29,39,30,38,28,23)(18,35,26,34,37,40,36,31)

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,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315806);