#487095

order	512
nilpotency class	4
isoclinism	isoclinic class of #486803
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	#29636	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

830 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394291);