#488325

order	512
nilpotency class	4
isoclinism	isoclinic class of #487691
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×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 763 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) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=gi,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=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, 395521);