#411199

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
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

656 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)(9,10,15,28,14,20,13,19)(11,21,23,32,24,17,29,16)(12,18,30,39,27,36,26,35)(22,34,37,40,25,33,38,31);(1,3,4,7)(2,5,6,8)(9,11)(10,16,20,32)(12,22)(13,23)(14,24)(15,29)(17,19,21,28)(18,31,36,40)(25,27)(26,37)(30,38)(33,35,34,39);(9,11)(10,17,20,21)(12,25)(13,23)(14,24)(15,29)(16,28,32,19)(18,34,36,33)(22,27)(26,38)(30,37)(31,35,40,39);(2,6)(5,8)(9,12,13,26,14,27,15,30)(10,18,19,35,20,36,28,39)(11,22,23,37,24,25,29,38)(16,31,21,34,32,40,17,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265390, #72307302, #72307361
maximal normal subgroup of	#72265390, #72307302, #72307361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318395);