#483398

order	512
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#106
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

495 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287549, #72287576, #72287594, …
maximal normal subgroup of	#72287549, #72287576, #72287594, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390594);