#484845

order	512
nilpotency class	4
isoclinism	isoclinic class of #484587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

767 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 767 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392041);