#496284

order	512
nilpotency class	4
isoclinism	isoclinic class of #481393
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#3061	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1036 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)(3,7)(4,6)(5,8)(9,10,12,17,14,20,26,19)(11,15,23,33,24,34,21,31)(13,27,25,18,29,30,38,37)(16,28,35,22,36,39,32,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,23)(13,22)(14,24)(16,30)(17,33)(18,32)(19,31)(20,34)(21,26)(25,39)(27,36)(28,38)(29,40)(35,37);(3,8)(5,7)(9,11)(10,16)(12,21)(13,28)(14,24)(15,30)(17,32)(18,33)(19,35)(20,36)(22,38)(23,26)(25,40)(27,34)(29,39)(31,37);(2,6)(5,7)(9,12,14,26)(10,17,20,19)(11,21,24,23)(13,25,29,38)(15,31,34,33)(16,32,36,35)(18,30,37,27)(22,28,40,39)

Polycyclic presentation:

2^9|a^2=g,d^2=i,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	#72343246, #72343252, #72343264
maximal normal subgroup of	#72343246, #72343252, #72343264
maximal quotient of	#72343721

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403480);