#511843

order	512
nilpotency class	4
isoclinism	isoclinic class of #511843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2118 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)(7,13)(10,19)(14,26)(15,27)(16,23)(17,22)(18,28)(20,32)(21,29)(24,31)(25,30);(1,3)(2,7)(4,13)(5,18)(6,14)(8,20)(9,12)(10,25)(11,21)(15,19)(16,29)(17,32)(22,28)(23,26)(24,27)(30,31);(9,23)(12,26)(22,24)(27,28);(2,8,11,10)(4,16)(7,20,21,25)(9,24,23,22)(12,27,26,28)(13,29)(17,31)(30,32)

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,f^b=fhi,g^b=gi,e^c=eh,e^d=ei,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260522
maximal normal subgroup of	#72260522
maximal quotient of	#72261166, #72284391, #72285658

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419039);