#106178

order	512
nilpotency class	3
isoclinism	isoclinic class of #8359
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1017 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=dh,e^b=eh,d^c=dh,e^c=eh,f^c=fhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285816, #72342987, #72342992, …
maximal normal subgroup of	#72285816, #72342987, #72342992, …
maximal quotient of	#72285992

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13374);