#488334

order	512
nilpotency class	4
isoclinism	isoclinic class of #487723
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	C₂²×D₄	C₂×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

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

normal subgroups only

This group has 742 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,7,4)(3,11,14,16)(5,10,6,9)(8,15,12,13)(17,18,23,29)(19,28,22,26)(20,31,30,24)(21,25,32,27);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,11)(10,16)(17,19,23,22)(18,24,29,31)(20,32,30,21)(25,28,27,26);(1,4,7,2)(3,12,14,8)(5,9,6,10)(11,15,16,13)(17,20,23,30)(18,25,29,27)(19,21,22,32)(24,26,31,28);(1,5,7,6)(2,9,4,10)(3,13,14,15)(8,11,12,16)(17,19,23,22)(18,26,29,28)(20,21,30,32)(24,25,31,27)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,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,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,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, 395530);