#137396

order	512
nilpotency class	4
isoclinism	isoclinic class of #12117
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

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

normal subgroups only

This group has 574 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)(5,6,8,14,25,17,27,15,11,18,26,30,10,16,9,12)(7,19,20,34,28,29,21,32,23,35,36,13,22,31,24,33);(1,2,3,4)(6,12)(8,9)(10,25)(13,29)(14,16)(15,18)(17,30)(19,33)(20,24)(21,36)(22,28)(26,27)(31,34)(32,35);(5,7,11,23)(6,13,18,34)(8,24,26,21)(9,20,27,36)(10,28,25,22)(12,29,15,31)(14,19,30,35)(16,33,17,32)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=gi,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eghi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,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	#72250968, #72256999, #72257005, …
maximal normal subgroup of	#72250968, #72256999, #72257005, …
maximal quotient of	#72265549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44592);