#137446

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	C₂×#469	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

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

normal subgroups only

This group has 402 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,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	#72250952, #72250955, #72250961, …
maximal normal subgroup of	#72250952, #72250955, #72250961, …
maximal quotient of	#72265406, #72265408

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44642);