#441634

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

917 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348830);