#486090

order	512
nilpotency class	4
isoclinism	isoclinic class of #486086
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18186
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#12173	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

655 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 655 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231715, #72307849, C₂×#486090
maximal normal subgroup of	#72231715, #72307849, C₂×#486090
maximal quotient of	#72250617, #72257006, C₂×#486090, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393286);