#146990

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#102
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210412, #72210551
maximal normal subgroup of	#72210412, #72210551
maximal quotient of	#72210378, #72210445

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54186);