#6454049

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437621
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3265 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,25)(20,26)(21,27)(22,29)(23,30)(24,32)(28,34)(31,35)(33,37)(36,39)(38,40);(9,11)(10,15)(12,19)(13,20)(14,21)(16,28)(17,31)(18,27)(22,33)(23,34)(24,35)(25,30)(26,32)(29,37)(36,40)(38,39);(2,5)(6,8)(9,12,14,23)(10,16,18,30)(11,19,21,34)(13,22,24,36)(15,25,27,28)(17,29,32,39)(20,33,35,40)(26,37,31,38);(9,13)(10,17)(11,20)(12,22)(14,24)(15,26)(16,29)(18,32)(19,33)(21,35)(23,36)(25,37)(27,31)(28,38)(30,39)(34,40);(1,3,4,7)(2,6,5,8)(11,21)(15,27)(19,34)(20,35)(25,28)(26,31)(33,40)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257536, #72279425, #72281069, …
maximal normal subgroup of	#72257536, #72279425, #72281069, …
maximal quotient of	#72264492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361245);