#486714

order	512
nilpotency class	4
isoclinism	isoclinic class of #486299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

785 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 785 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(6,12)(13,16)(17,18)(19,29)(20,25)(21,34)(22,38)(23,28)(24,31)(26,33)(27,36)(30,45)(32,41)(35,40)(37,44)(39,46)(42,48)(43,47);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,24)(20,30)(21,35)(22,39)(23,32)(25,40)(26,45)(27,46)(28,43)(29,34)(31,36)(33,47)(37,48)(38,42)(41,44);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20,23,22)(18,25,28,38)(19,30,32,39)(21,33,37,36)(24,40,43,42)(26,44,27,34)(29,45,41,46)(31,35,47,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,20,33,23,37,22,36)(18,26,38,34,28,27,25,44)(19,31,39,48,32,47,30,35)(24,41,40,46,43,29,42,45)

Polycyclic presentation:

2^9|c^2=i,d^2=ghi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265737, #72308622
maximal normal subgroup of	#72265737, #72308622

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393910);