#144866

order	512
nilpotency class	4
isoclinism	isoclinic class of #142216
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

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	C₂×#392	C₂²
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

797 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,e^2=g,f^2=gi,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,c^b=cf,e^b=eh,f^b=fgi,e^c=eg,f^c=fgi,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, 52062);