#142863

order	512
nilpotency class	4
isoclinism	isoclinic class of #142220
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×C₄⋊C₄	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 621 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6,9,13)(7,14,17,15)(8,19,10,12)(11,18,20,16)(22,26)(27,33)(28,34)(35,36);(5,7,9,17)(6,11,13,20)(8,18,10,16)(12,15,19,14)(21,22)(23,27)(24,26)(25,28)(29,33)(30,35)(31,36)(32,34);(2,4)(5,6)(7,15)(8,12)(9,13)(10,19)(11,16)(14,17)(18,20)(21,23,25,30)(22,27,28,35)(24,31,32,29)(26,33,34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,f^b=fg,h^b=hi,d^c=dh,f^c=fg,f^d=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, 50059);