#511844

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

Polycyclic presentation:

2^9|c^2=i,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, #72260545, #72284822, …
maximal normal subgroup of	#72260522, #72260545, #72304193
maximal quotient of	#72261195, #72261263, #72262574, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419040);