#137401

order	512
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
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)(5,6,9,15,10,17,27,30,12,18,8,16,11,29,25,13)(7,19,21,36,22,35,26,14,24,31,20,32,23,34,28,33);(1,2,3,4)(6,13,18,30)(8,25)(9,27)(10,11)(14,19,33,31)(15,17,16,29)(20,28)(21,26)(22,23)(32,34,36,35);(5,7,12,24)(6,14,18,33)(8,26,9,28)(10,23,11,22)(13,19,30,31)(15,34,16,35)(17,36,29,32)(20,25,21,27)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,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	#72250965, #72256999, #72257006, …
maximal normal subgroup of	#72250965, #72256999, #72257006, …
maximal quotient of	#72265549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44597);