#6447742

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3094 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318987, #72319059, #72319377, …
maximal normal subgroup of	#72318987, #72319059, #72319377, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354938);