#403745

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1026 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280196
maximal normal subgroup of	#72280196

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310941);