#496167

order	512
nilpotency class	4
isoclinism	isoclinic class of #483956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

479 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)(3,8)(4,6)(5,7)(9,10,12,15)(11,19)(13,26,24,40)(14,28,25,18)(16,31)(17,34,30,27)(20,36)(21,35)(22,29)(23,39)(32,37)(33,38);(1,3,5,4)(2,6,7,8)(10,15)(16,29)(17,30)(18,28)(19,35)(26,40)(32,33)(36,39);(9,11,13,22,14,23,27,38,12,21,24,31,25,20,34,37)(10,16,17,19,18,33,26,36,15,29,30,35,28,32,40,39);(1,4,5,3)(2,6,7,8)(11,20,21,23)(13,24)(16,32,29,33)(17,30)(19,36,35,39)(22,38,31,37)(26,40)(27,34)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=f,d^2=eg,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309671, #72310194, #72310206, …
maximal normal subgroup of	#72309671, #72310194, #72310206, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403363);