#145823

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	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

651 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 651 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=dgh,e^2=gi,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fi,d^c=di,e^c=egi,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, 53019);