#496005

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
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	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 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)(11,19,21,37)(12,23)(13,26)(15,27,29,39)(17,32)(18,31)(20,36,22,24)(25,35)(28,33,34,40)(30,38);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,27)(20,34)(21,29)(22,28)(23,32)(24,33)(25,30)(26,31)(35,38)(36,40)(37,39);(9,11)(10,15)(12,24)(13,20)(14,21)(16,29)(17,33)(18,34)(19,35)(22,26)(23,37)(25,36)(27,38)(28,31)(30,40)(32,39);(1,4,5,3)(2,6,7,8)(10,16)(11,20,21,22)(12,25)(15,28,29,34)(18,31)(19,36,37,24)(23,35)(27,33,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313244, #72313282, #72313292, …
maximal normal subgroup of	#72313244, #72313282, #72313292, …
maximal quotient of	#72343348, #72343424

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403201);