#496373

order	512
nilpotency class	4
isoclinism	isoclinic class of #481393
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265445, #72265458, #72265521, …
maximal normal subgroup of	#72265445, #72265458, #72265521, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403569);