#408192

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1003 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)(11,21)(12,25)(13,20)(14,18)(15,19)(16,24)(17,27)(22,36)(23,34)(26,33)(28,32)(29,30)(31,37)(35,38)(39,40);(1,3,5,4)(2,6,7,8)(16,30)(21,34)(31,36)(35,40);(9,11)(10,16)(12,22)(13,29)(14,24)(15,23)(17,31)(18,34)(19,30)(20,21)(25,35)(26,39)(27,38)(28,37)(32,40)(33,36);(1,4,5,3)(2,6,7,8)(9,12,13,26,15,28,14,27)(10,17,18,32,19,33,20,25)(11,22,29,39,23,37,24,38)(16,31,34,40,30,36,21,35)

Polycyclic presentation:

2^9|b^2=efh,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^c=ei,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, 315388);