#410069

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

1152 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)(3,6)(4,5)(7,8)(9,10)(11,23)(12,28)(13,32)(14,22)(15,20)(16,21)(17,25)(18,30)(19,24)(26,36)(27,35)(29,38)(31,39)(33,40)(34,37);(2,5)(6,8)(9,11,15,27,16,26,14,25)(10,17,22,36,21,35,20,23)(12,24,31,34,30,33,29,13)(18,37,39,19,28,32,38,40);(9,12)(10,18,21,28)(11,24)(13,25)(14,29)(15,31)(16,30)(17,32,35,37)(19,36,40,23)(20,38,22,39)(26,33)(27,34);(1,3,4,7)(2,6,5,8)(9,13,14,33,16,34,15,24)(10,19,20,37,21,40,22,32)(11,12,25,29,26,30,27,31)(17,28,23,39,35,18,36,38)

Polycyclic presentation:

2^9|b^2=h,c^2=g,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,f^c=fi

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, 317265);