#410070

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

1138 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)(11,21)(12,26)(13,29)(14,32)(15,20)(16,24)(17,23)(18,30)(19,33)(22,38)(25,36)(27,34)(28,37)(31,35)(39,40);(2,5)(6,8)(9,11,14,24,15,25,33,27)(10,16,19,21,20,34,32,36)(12,22,28,13,23,30,40,31)(17,35,37,18,26,29,39,38);(9,12)(10,17,20,26)(11,22)(13,24)(14,28)(15,23)(16,29,34,35)(18,21,38,36)(19,37,32,39)(25,30)(27,31)(33,40);(1,3,4,7)(2,6,5,8)(9,13,14,30,15,31,33,22)(10,18,19,29,20,38,32,35)(11,23,24,40,25,12,27,28)(16,26,21,39,34,17,36,37)

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,d^c=di,f^c=fi

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, 317266);