#6470045

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4130 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)(10,16)(13,14)(17,32)(18,31)(19,28)(20,30)(22,29)(24,34)(25,27)(33,40)(35,39)(36,37);(1,3)(2,5)(4,7)(6,8)(9,10)(11,20)(12,24)(13,16)(14,28)(15,19)(17,23)(18,26)(21,33)(22,32)(25,31)(27,34)(29,30)(35,38)(36,40)(37,39);(3,7)(5,8)(9,11,15,23)(10,17,19,20)(12,21,26,38)(13,22,14,29)(16,30,28,32)(18,33,24,35)(25,36,27,37)(31,39,34,40);(3,7)(5,8)(9,12)(10,18)(11,21)(13,27)(14,25)(15,26)(16,31)(17,33)(19,24)(20,35)(22,37)(23,38)(28,34)(29,36)(30,39)(32,40);(2,6)(5,8)(9,13,15,14)(10,16,19,28)(11,22,23,29)(12,25,26,27)(17,30,20,32)(18,34,24,31)(21,36,38,37)(33,40,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318287
maximal normal subgroup of	#72318287

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377241);