#6446718

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286920, #72286939, #72286946, …
maximal normal subgroup of	#72286920, #72286939, #72286946, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353914);