#511841

order	512
nilpotency class	4
isoclinism	isoclinic class of #511841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#32899
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2123 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260481, #72306736
maximal normal subgroup of	#72260481, #72306736
maximal quotient of	#72261096

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419037);