#488028

order	512
nilpotency class	4
isoclinism	isoclinic class of #487707
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	C₂×#127	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

761 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395224);