#192926

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1103 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=eh,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100122);