#146537

order	512
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1823
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

986 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231484, #72231520, #72231521, …
maximal normal subgroup of	#72231484, #72231520, #72231521, …
maximal quotient of	#72265309, #72280500, C₂×#146537

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53733);