#147093

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
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	C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

574 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)(3,4)(5,6,7,8)(9,10)(11,18)(12,27)(13,30)(14,33)(15,34)(16,22)(17,23)(19,32)(20,28)(21,31)(24,36)(25,40)(26,35)(29,37)(38,39);(2,4)(9,11,14,24,31,39,32,29,16,26,13,23,15,25,28,12)(10,17,19,18,34,37,33,40,22,36,20,35,21,27,30,38);(9,12,28,25,15,23,13,26,16,29,32,39,31,24,14,11)(10,18,33,36,21,38,19,37,22,35,30,17,34,40,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210412
maximal normal subgroup of	#72210412
maximal quotient of	#72210358

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54289);