#205648

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

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

1328 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)(9,10,14,20,15,19,13,18)(11,16,22,30,23,32,28,31)(12,24,26,34,27,33,25,17)(21,35,36,29,37,39,38,40);(1,2,6,8,4,7,3,5)(9,11,13,22,15,23,14,28)(10,16,20,31,19,32,18,30)(12,21,25,36,27,37,26,38)(17,29,24,35,34,40,33,39);(9,12,13,25,15,27,14,26)(10,17,20,24,19,34,18,33)(11,21,22,36,23,37,28,38)(16,29,31,35,32,40,30,39);(9,13,15,14)(10,18,19,20)(11,22,23,28)(12,25,27,26)(16,30,32,31)(17,33,34,24)(21,36,37,38)(29,39,40,35)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=i,e^2=hi,f^2=i,g^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, #72224860, #72224863, …
maximal normal subgroup of	#72224856, #72224860, #72224863, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112844);