#6437606

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237505, C₂×#6437606
maximal normal subgroup of	#72237505, C₂×#6437606
maximal quotient of	#72310300, #72310301, C₂×#6437606

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344802);