#144764

order	512
nilpotency class	4
isoclinism	isoclinic class of #142216
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

589 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=dg,e^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,c^b=cf,e^b=eh,e^c=eg,h^c=hi,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, 51960);