#486037

order	512
nilpotency class	4
isoclinism	isoclinic class of #485643
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
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

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

normal subgroups only

This group has 841 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,16)(13,15)(17,18)(19,22)(20,23)(21,26)(24,27)(25,29)(28,31)(30,32);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,16)(10,14)(18,22)(19,27)(20,29)(23,31)(24,26)(25,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(18,23)(20,29)(22,31)(24,32)(25,26)(28,30);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)

Polycyclic presentation:

2^9|c^2=gi,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=dgi,e^b=eg,f^b=fhi,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, 393233);