#411196

order	512
nilpotency class	3
isoclinism	isoclinic class of #394746
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

670 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)(3,5)(4,6)(7,8)(9,10,14,19)(11,20)(12,17,26,33)(13,16,27,18)(15,23)(21,34)(22,30)(24,28)(25,31,38,32)(29,36)(35,40)(37,39);(1,3,4,7)(2,5,6,8)(9,11)(10,15,19,30)(12,21)(13,22)(14,24)(16,20,18,28)(17,29,33,40)(23,27)(25,35)(26,37)(31,34,32,39)(36,38);(10,16,19,18)(15,28,30,20)(17,31,33,32)(29,39,40,34);(2,6)(5,8)(9,12,13,25,14,26,27,38)(10,17,16,31,19,33,18,32)(11,21,22,35,24,37,23,36)(15,29,20,34,30,40,28,39)

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=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, 318392);