#488858

order	512
nilpotency class	4
isoclinism	isoclinic class of #488719
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	#3169	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 645 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=gh,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,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, 396054);