#487066

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	#18152	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

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

normal subgroups only

This group has 802 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)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,43)(32,44)(33,39)(34,48)(35,41)(36,42)(37,45)(38,47)(40,46);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19,23,32)(18,24,28,38)(20,30,21,34)(22,31,36,47)(25,37,26,40)(27,29,42,44)(33,45,35,46)(39,43,41,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(17,20,23,21)(18,25,28,26)(19,30,32,34)(22,35,36,33)(24,37,38,40)(27,39,42,41)(29,43,44,48)(31,46,47,45);(3,9)(7,14)(10,15)(13,16)(17,21,23,20)(18,26,28,25)(19,30,32,34)(22,35,36,33)(24,37,38,40)(27,41,42,39)(29,43,44,48)(31,45,47,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,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=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, 394262);