#144874

order	512
nilpotency class	4
isoclinism	isoclinic class of #142258
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#30253	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

464 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64

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

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

Polycyclic presentation:

2^9|c^2=d,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,g^b=gi,h^b=hi,e^c=eg,f^c=fg,g^c=gi,f^e=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, 52070);