#137393

order	512
nilpotency class	4
isoclinism	isoclinic class of #12114
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 602 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) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=gi,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=egh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,g^c=gi,h^c=hi,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250965, #72250970, #72257005, …
maximal normal subgroup of	#72250965, #72250970, #72257005, …
maximal quotient of	#72265567

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44589);