#137447

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 374 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)(5,6,10,14,12,28,9,26,13,18,25,30,11,17,8,16)(7,19,21,15,23,31,36,35,24,29,27,33,22,34,20,32);(1,2,3,4)(6,14,18,30)(8,25)(9,10)(11,12)(15,29,33,19)(16,17,26,28)(20,27)(21,36)(22,23)(31,32,34,35);(5,7,13,24)(6,15,18,33)(8,21,9,27)(10,20,25,36)(11,23,12,22)(14,29,30,19)(16,34,26,31)(17,35,28,32)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=i,d^2=g,e^2=gi,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=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	#72250955, #72256999, #72257006, …
maximal normal subgroup of	#72250955, #72256999, #72257006, …
maximal quotient of	#72265546

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44643);