#6347124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347092
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3955 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,31)(20,27)(21,26)(22,29)(23,28)(24,30)(25,33)(32,39)(34,37)(35,38)(36,40);(2,5)(3,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,25)(17,26)(18,27)(22,32)(23,33)(24,34)(28,31)(29,36)(30,37)(35,40)(38,39);(9,12)(10,16)(11,19)(13,22)(14,23)(15,25)(17,28)(18,29)(20,32)(21,33)(24,35)(26,31)(27,36)(30,38)(34,40)(37,39);(10,17)(15,26)(16,28)(18,30)(25,31)(27,37)(29,38)(36,39);(2,6)(5,7)(9,13,14,24)(10,18,17,30)(11,20,21,34)(12,22,23,35)(15,27,26,37)(16,29,28,38)(19,32,33,40)(25,36,31,39)

Polycyclic presentation:

2^9|e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,f^b=fh,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279425, #72281484, #72281704, …
maximal normal subgroup of	#72279425, #72281484, #72281704, …
maximal quotient of	#72282752

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254320);