#511786

order	512
nilpotency class	4
isoclinism	isoclinic class of #511785
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,d^b=di,f^b=fhi,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234427, #72261685, #72306753, …
maximal normal subgroup of	#72234427, #72261685, #72306753, …
maximal quotient of	#72233115

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418982);