#6451745

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3938 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)(3,8)(4,6)(5,7)(9,10)(11,20)(12,17)(13,18)(14,27)(15,19)(16,23)(21,37)(22,38)(24,32)(25,34)(26,36)(28,33)(29,35)(30,39)(31,40);(9,11)(10,16)(12,21)(13,23)(14,28)(15,24)(17,31)(18,32)(19,33)(20,27)(22,26)(25,39)(29,40)(30,36)(34,37)(35,38);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,34)(19,36)(20,37)(22,24)(23,39)(27,35)(28,40)(31,33)(32,38);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,25)(14,29)(15,26)(16,31)(18,34)(19,36)(20,38)(21,24)(23,40)(27,35)(28,39)(30,33)(32,37);(9,13,15,14)(10,18,19,27)(11,23,24,28)(12,25,26,29)(16,32,33,20)(17,34,36,35)(21,39,22,40)(30,38,31,37)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258209, #72258210, #72267952, …
maximal normal subgroup of	#72258209, #72258210, #72267952, …
maximal quotient of	#72258248

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358941);