#486680

order	512
nilpotency class	4
isoclinism	isoclinic class of #486320
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
prime radical	#62270	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 742 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393876);