#142806

order	512
nilpotency class	4
isoclinism	isoclinic class of #142256
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

654 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 72 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)(5,6,13,8)(7,19,23,33)(9,26,28,16)(10,25,27,15)(11,18,12,17)(14,29,24,20)(21,34,35,32)(22,36,30,31);(5,7)(6,14)(8,24)(9,21)(10,29)(11,30)(12,22)(13,23)(15,31)(16,33)(17,34)(18,32)(19,26)(20,27)(25,36)(28,35);(2,4)(5,8,13,6)(7,19,23,33)(9,26,28,16)(10,25,27,15)(11,17,12,18)(14,29,24,20)(21,34,35,32)(22,36,30,31)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,d^c=dh,f^c=fg,g^c=gi,h^c=hi,f^d=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, 50002);