#137452

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 380 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)(5,6,10,13,11,17,8,25,12,19,9,16,27,18,28,15)(7,20,22,35,23,32,21,14,24,33,26,34,30,31,29,36);(1,2,3,4)(6,13)(8,9)(10,28)(11,27)(14,31)(15,17)(16,19)(18,25)(20,35)(21,26)(22,29)(23,30)(32,36)(33,34);(5,7,12,24)(6,14,19,36)(8,26,28,22)(9,21,10,29)(11,30,27,23)(13,31,16,32)(15,20,25,33)(17,35,18,34)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=i,d^2=gi,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egh,h^a=hi,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fhi,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	#72250959, #72256999, #72257005, …
maximal normal subgroup of	#72250959, #72256999, #72257005, …
maximal quotient of	#72265546

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44648);