#205537

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1348 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224856, #72224858, #72224861, …
maximal normal subgroup of	#72224856, #72224858, #72224861, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112733);