#6516749

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499997
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	#62824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3352 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268466, #72268468, #72268479
maximal normal subgroup of	#72268466, #72268468, #72268479

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6423945);