#486785

order	512
nilpotency class	4
isoclinism	isoclinic class of #486327
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

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

normal subgroups only

This group has 671 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=gh,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393981);