#485382

order	512
nilpotency class	4
isoclinism	isoclinic class of #485343
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×C₄⋊C₄	C₂⁴
prime radical	#3493	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,24)(20,31)(21,26)(22,36)(23,28)(25,33)(27,37)(29,43)(30,42)(32,46)(34,45)(35,39)(38,44)(40,47)(41,48);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,21)(18,24,28,26)(20,29,32,34)(22,30,37,35)(25,38,40,41)(27,42,36,39)(31,45,46,43)(33,44,47,48);(17,20)(18,25,28,40)(19,29)(21,34)(22,33)(23,32)(24,38,26,41)(27,31,36,46)(30,44)(35,48)(37,47)(39,43,42,45);(2,7)(6,15)(8,12)(9,16)(17,21,23,19)(18,26,28,24)(20,29,32,34)(22,35,37,30)(25,38,40,41)(27,42,36,39)(31,43,46,45)(33,44,47,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=di,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, 392578);