#410072

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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1104 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,26)(13,17)(14,19)(15,20)(16,23)(18,29)(22,30)(24,32)(25,31)(27,33)(28,34)(35,38)(36,40)(37,39);(2,5)(6,8)(9,11,14,24,15,25,13,23)(10,16,17,31,20,32,19,21)(12,22,29,38,27,36,28,37)(18,30,26,39,34,40,33,35);(10,17,20,19)(12,27)(16,21,32,31)(18,33,34,26)(22,36)(28,29)(30,39,40,35)(37,38);(1,3,4,7)(2,6,5,8)(9,12,13,28,15,27,14,29)(10,18,19,33,20,34,17,26)(11,22,23,37,25,36,24,38)(16,30,21,35,32,40,31,39)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=e,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,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, 317268);